*4.2. Pile–Pile Interaction Factor*

The regressive solution (the pile is considered completely embedded in the soil) of the proposed calculation model is compared with the solution of Kaynia [36]. In this section, parameters below are used for calculation: the ratio of elastic modulus of the pile to elastic modulus of the soil *Ep*/*Es* = 1000, the ratio of the pile length to the pile diameter

*L*/*D* = 20, and Poisson's ratio is 0.4. The ratio of the density of the pile to the density of the soil *ρp*/*ρ<sup>s</sup>* = 1.3, while the damping ratio *β<sup>s</sup>* = 0.05. The pile spacing *S* = 2*D*, the pile embedment ratio *Lw*/*L* = 0.3. The dimensionless frequency *a*<sup>0</sup> = *ωD*/*Vs* is used for analysis. Here, we consider the influence of three parameters on the pile–pile interaction factor: pile spacing, pile embedment ratio, and the angle of incidence *θ*.

As shown in Figure 12, when the pile spacing *S*/*D* = 10, the calculating result of this paper and the result from the literature are in good agreement, which validates the correctness of the calculation method. When the pile spacing *S*/*D* = 5, some differences can be observed compared with the results from the literature. Furthermore, when *S*/*D* = 2, the difference becomes more significant, especially in low frequencies. This is because when the pile spacing decreases, the influence of the passive pile on the active pile is more significant, which changes the horizontal displacement of the active pile. This analysis demonstrates that when the pile spacing is small, the influence of the passive pile on the active pile is important and cannot be ignored.

**Figure 12.** Pile–pile interaction under different pile spacings.

The influence of the pile embedded ratio is studied. As shown in Figure 13, when the pile is completely embedded, the pile–pile interaction factor decreases with increasing load frequency, and the value remains positive. When *Lw*/*L* = 0.1, the pile–pile interaction factor *α* at all frequencies largely decreases. When *Lw*/*L* = 0.2, *α* changes obviously. With increasing load frequency, *α* decreases rapidly when closing to a certain frequency and rises after reaching the lowest point. When *Lw*/*L* = 0.3, the curve moves from a mainly middle-high frequency to a middle-low frequency. This analysis demonstrates that *α* is sensitive to the pile embedment ratio. When the pile embedment ratio is relatively small, the change of *α* with increasing frequency is simple; when the pile embedment ratio is relatively large, the change of *α* becomes complex.

**Figure 13.** Pile–pile interaction under different pile embedded ratios.

The influence of different incidences angle *θ* is studied, i.e., 0◦ , 45◦ , and 90◦ , respectively. As shown in Figure 14, when *θ* = 0 ◦ , the curve fluctuates significantly, the minimum value of *α* is the smallest among three angles, and the maximum value of *α* is the largest among three angles. When *θ* = 90◦ , the value of *α* is slightly smaller than that of *θ* = 0 ◦ . When *θ* = 45◦ , the change of *α* is moderate, and no significant fluctuation is observed.

**Figure 14.** Pile–pile interaction under different incidence angles.

#### *4.3. Dynamic Foundation Impedance*

As shown in Figure 15, with increasing load frequency, the change of horizonal foundation impedance is not obvious. When the load frequency is around soil cut-off frequency, the horizontal foundation impedance slightly decreases. The foundation impedance is very sensitive to the pile length submerged in the seawater. With an increasing pile embedment ratio, the foundation impedance decreases significantly.

**Figure 15.** The influence of the pile length in seawater on the foundation impedance.

Meanwhile, the wave load can influence the foundation impedance to a certain degree. As shown in Figure 16, when the wave load is relatively small (wavelength *Lwl* = 75 m, wave height *Hw* = 4 m), the influence of wave load on the foundation impedance is not significant. When the wave load is relatively large (wavelength *Lwl* = 120 m, wave height *Hw* = 8 m), the influence of wave load on the foundation impedance is significant.

**Figure 16.** The influence of the wave load on the foundation impedance.

#### **5. Conclusions**

This paper establishes the model of pile group supported OWT under wind and wave load; the main findings are presented below:


**Author Contributions:** Conceptualization, W.Y.; Data curation, Y.S.; Formal analysis, Y.S.; Funding acquisition, W.Y.; Investigation, G.Y.; Methodology, Y.S.; Project administration, G.Y.; Resources, G.Y.; Software, Y.S.; Supervision, W.Y.; Validation, Y.S.; Visualization, Y.S.; Writing—original draft, Y.S.; Writing—review and editing, Y.S. and W.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This study was financially supported by the Key Projects of the National Natural Science Foundation of China (Grant No. 11932010).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

$$\{f\_{i}^{11}\} = \begin{pmatrix} e^{(-1+i)\gamma z} & e^{(-1-i)\gamma z} & e^{(1+i)\gamma z} & e^{(1+i)\gamma z} & q\_{f\_{G}}^{\prime}(z) \\ (-1+i)\gamma e^{(-1+i)\gamma z} & (-1-i)\gamma e^{(-1-i)\gamma z} & (1-i)\gamma e^{(1-i)\gamma z} & (1+i)\gamma e^{(1+i)\gamma z} & K \frac{\sinh(\mathcal{K}z)}{\cosh(\mathcal{K}z)} q\_{,G}^{\prime}(z) \\ (-1+i)^{2}\gamma^{2} e^{(-1+i)\gamma z} & (-1-i)^{2}\gamma^{2} e^{(-1-i)\gamma z} & (1-i)^{2}\gamma^{2} e^{(1-i)\gamma z} & (1+i)^{2}\gamma^{2} e^{(1+i)\gamma z} & K^{2} q\_{,G}^{\prime}(z) \\ (-1+i)^{3}\gamma^{3} e^{(-1+i)\gamma z} & (-1-i)^{3}\gamma^{3} e^{(-1-i)\gamma z} & (1-i)^{3}\gamma^{3} e^{(1-i)\gamma z} & (1+i)^{3}\gamma^{3} e^{(1+i)\gamma z} & K^{3} \frac{\sinh(\mathcal{K}z)}{\cosh(\mathcal{K}z)} q\_{,G}^{\prime}(z) \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}$$

$$
\begin{Bmatrix}
\mathfrak{e}^{\ell(-1+i)\gamma z}\_{i} & \mathfrak{e}^{(-1-i)\gamma z}\_{i} & \mathfrak{e}^{(1-i)\gamma z}\_{i} & \mathfrak{e}^{(1+i)\gamma z}\_{i} & 0 \\
\{t^{2}\_{i}\} & (-1+i)\gamma \mathfrak{e}^{(-1+i)\gamma z}\_{i} & (-1-i)\gamma \mathfrak{e}^{(-1-i)\gamma z}\_{i} & (1-i)\gamma \mathfrak{e}^{(1-i)\gamma z}\_{i} & (1+i)\gamma \mathfrak{e}^{(1+i)\gamma z}\_{i} & 0 \\
(-1+i)^{2}\gamma^{2} \mathfrak{e}^{(-1+i)\gamma z}\_{i} & (-1-i)^{2}\gamma^{2} \mathfrak{e}^{(-1-i)\gamma z}\_{i} & (1-i)^{2}\gamma^{2} \mathfrak{e}^{(1-i)\gamma z}\_{i} & (1+i)^{2}\gamma^{2} \mathfrak{e}^{(1+i)\gamma z}\_{i} & 0 \\
(-1+i)^{3}\gamma^{3} \mathfrak{e}^{(-1+i)\gamma z}\_{i} & (-1-i)^{3}\gamma^{3} \mathfrak{e}^{(-1-i)\gamma z}\_{i} & (1-i)^{3}\gamma^{3} \mathfrak{e}^{(1-i)\gamma z}\_{i} & (1+i)^{3}\gamma^{3} \mathfrak{e}^{(1+i)\gamma z}\_{i} & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
$$

$$\begin{Bmatrix} t\_i^p \end{Bmatrix} = \begin{Bmatrix} z(1-i)e^{(-1+i)\gamma z} & z(1+i)e^{(-1-i)\gamma z} \\ (1-i)e^{(-1+i)\gamma z} + z(2i\gamma)e^{(-1+i)\gamma z} & (1+i)e^{(-1-i)\gamma z} + z(-2i\gamma)e^{(-1-i)\gamma z} \\ (4i\gamma)e^{(-1+i)\gamma z} - 2z(i+1)\gamma^2 e^{(-1+i)\gamma z} & (-4i\gamma)e^{(-1-i)\gamma z} + 2z(i-1)\gamma^2 e^{(-1-i)\gamma z} \\ -6(i+1)\gamma^2 e^{(-1+i)\gamma z} + 4z\gamma^3 e^{(-1+i)\gamma z} & 6(i-1)\gamma^2 e^{(-1-i)\gamma z} + 4z\gamma^3 e^{(-1-i)\gamma z} \\ 0 & 0 \\ z(-1+i)e^{(1-i)\gamma z} & z(-1-i)e^{(1+i)\gamma z} & 0 \\ (-1+i)e^{(1-i)\gamma z} + z(2i\gamma)e^{(-1-i)\gamma z} & (-1-i)e^{(1+i)\gamma z} + z(-2i\gamma)e^{(1+i)\gamma z} & 0 \\ (4i\gamma)e^{(-1-i)\gamma z} + 2z(i+1)\gamma^2 e^{(1-i)\gamma z} & (-4i\gamma)e^{(1+i)\gamma z} - 2(i-1)\gamma^2 z e^{(1+i)\gamma z} & 0 \\ 6(i+1)\gamma^2 e^{(1-i)\gamma z} + 4z\gamma^3 e^{(1-i)\gamma z} & -6(i-1)\gamma^2 e^{(1+i)\gamma z} + 4\gamma^3 z e^{(1+i)\gamma z} & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

 *t a*2 *i* = (*k*+*iωc*)*<sup>ψ</sup>* 512*Ep Ipγ*<sup>6</sup> ⎧ ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ <sup>−</sup>2*iz*2*e*(−1+*i*)*γ<sup>z</sup>* <sup>−</sup>4*ize*(−1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>2</sup>(*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)*γz*2*e*(−1+*i*)*γ<sup>z</sup>* <sup>−</sup>4*ie*(−1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>8</sup>(*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)*γze*(−1+*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>4</sup>*γ*2*z*2*e*(−1+*i*)*γ<sup>z</sup>* <sup>12</sup>(<sup>1</sup> <sup>+</sup> *<sup>i</sup>*)*γe*(−1+*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>24</sup>*γ*2*ze*(−1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>4</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γ*3*z*2*e*(−1+*i*)*γ<sup>z</sup>* 0 2*iz*2*e*(−1−*i*)*γ<sup>z</sup>* <sup>4</sup>*ize*(−1−*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>2</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γz*2*e*(−1−*i*)*γ<sup>z</sup>* <sup>4</sup>*ie*(−1−*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>8</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γze*(−1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>4</sup>*γ*2*z*2*e*(−1−*i*)*γ<sup>z</sup>* <sup>12</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γe*(−1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>24</sup>*γ*2*ze*(−1−*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>4</sup>(<sup>1</sup> <sup>+</sup> *<sup>i</sup>*)*γ*3*z*2*e*(−1−*i*)*γ<sup>z</sup>* 0 <sup>−</sup>2*iz*2*e*(1−*i*)*γ<sup>z</sup>* <sup>−</sup>4*ize*(1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>2</sup>(*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)*γz*2*e*(1−*i*)*γ<sup>z</sup>* <sup>−</sup>4*ie*(1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>8</sup>(*<sup>i</sup>* <sup>+</sup> <sup>1</sup>)*γze*(1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>4</sup>*γ*2*z*2*e*(1−*i*)*γ<sup>z</sup>* <sup>−</sup>12(<sup>1</sup> <sup>+</sup> *<sup>i</sup>*)*γe*(1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>24</sup>*γ*2*ze*(1−*i*)*γ<sup>z</sup>* <sup>−</sup> <sup>4</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γ*3*z*2*e*(1−*i*)*γ<sup>z</sup>* 0 2*iz*2*e*(1+*i*)*γ<sup>z</sup>* 0 <sup>−</sup>4*ize*(1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>2</sup>(*<sup>i</sup>* <sup>−</sup> <sup>1</sup>)*γz*2*e*(1+*i*)*γ<sup>z</sup>* <sup>0</sup> <sup>−</sup>4*ie*(1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>8</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γze*(1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>4</sup>*γ*2*z*2*e*(1+*i*)*γ<sup>z</sup>* <sup>0</sup> <sup>12</sup>(<sup>1</sup> <sup>−</sup> *<sup>i</sup>*)*γe*(1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>24</sup>*γ*2*ze*(1+*i*)*γ<sup>z</sup>* <sup>+</sup> <sup>4</sup>(<sup>1</sup> <sup>+</sup> *<sup>i</sup>*)*γ*3*z*2*e*(1+*i*)*γ<sup>z</sup>* <sup>0</sup> 0 1 ⎫ ⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎭

#### **References**


**Jian Song \*, Junying Chen, Yufei Wu and Lixiao Li \***

College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China

**\*** Correspondence: jiansong@szu.edu.cn (J.S.); llxiao2021@gmail.com (L.L.)

**Abstract:** With the increase in wind turbine power, the size of the blades is significantly increasing to over 100 m. It is becoming more and more important to optimize the design for the internal layout of large-scale offshore composite wind turbine blades to meet the structural safety requirements while improving the blade power generation efficiency and achieving light weight. In this work, the full-scale internal layout of an NREL 5 MW offshore composite wind turbine blade is elaborately designed via the topology optimization method. The aerodynamic wind loads of the blades were first simulated based on the computational fluid dynamics. Afterwards, the variable density topology optimization method was adopted to perform the internal structure design of the blade. Then, the first and second generation multi-web internal layouts of the blade were reversely designed and evaluated in accordance with the stress level, maximum displacement of blade tip and fatigue life. In contrast with the reference blade, the overall weight of the optimized blade was reduced by 9.88% with the requirements of stress and fatigue life, indicating a better power efficiency. Finally, the vibration modal and full life cycle of the designed blade were analyzed. The design conception and new architecture could be useful for the improvement of advanced wind turbines.

**Keywords:** offshore wind turbine blade; composites; computational fluid dynamics; topology optimization; fatigue life
