**1. Introduction**

Floating platforms have been used for offshore oil and gas rigging, offshore renewable energy farms, aquaculture farms, floating hotels, floating parks, floating houses and floating entertainment/leisure facilities [1–3]. Floating breakwaters create a sheltered sea space that allows safe operation and maintenance of floating solar farms [4], fish farming [5], ship harbouring [6–8], etc. By placing a floating platform within a floating ring breakwater, one may have a practical solution for operating the aforementioned activities in an open sea. For example, Figure 1 shows a conceptual design of a mega offshore floating fish farm surrounded by a hexagonal floating breakwater in an open ocean. The internal floating hexagonal platform houses the control centre, power production and storage facility, fish processing plant, offices, workers' quarters, etc. Alternatively, the floating ring structure may be designed to trap waves to create a high wave energy environment with the view to harvest wave energy using the piston-like internal floating cylinder, i.e., Wave Energy Converter (WEC) device [9,10].

Garrett [8] determined the wave motion inside a thin-walled bottomless harbour using an analytical method. Mavrakos [11] and Mavrakos [12] extended Garrett [8] study for thick-walled floating bottomless circular cylinders and solved the diffraction and radiation problems, respectively. Later, Mavrakos and Chatjigeorgiou [13] tackled the second-order waves for the same problem in order to improve the inaccuracy of the linear wave potential theory due to the trapped waves in the inner water basin resulting in the highly amplified resonant waves. For two concentric floating circular cylinders, Mavrakos [14] and Mavrakos [15] obtained the wave exciting forces and hydrodynamic coefficients. Mavrakos, et al. [16] addressed tightly moored two concentric floating circular

**Citation:** Park, J.C.; Wang, C.M. Hydrodynamic Behaviour of a Floating Polygonal Platform Centrally Placed within a Polygonal Ring Structure under Wave Action. *J. Mar. Sci. Eng.* **2022**, *10*, 1430. https: //doi.org/10.3390/jmse10101430

Academic Editors: Carlos Guedes Soares and Serge Sutulo

Received: 2 September 2022 Accepted: 20 September 2022 Published: 4 October 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

cylinders under first and second-order waves. Konispoliatis, Mazarakos and Mavrakos [10] presented analytical solutions for an array of Oscillating Water Column (OWC) devices. Each OWC device consists of concentric circular cylinders.

**Figure 1.** HEXAGON: a mega offshore floating fish farm: (**a**) plan view; (**b**) isometric view.

However, the existing formulations are mostly based on floating circular cylinders as it is analytically tractable in the cylindrical coordinate system. In practice, non-circular models are used for marine structure applications [17–19]. Recently, Park and Wang [20] and Park and Wang [21] respectively investigated the hydrodynamic behaviours of floating polygonal platforms and floating polygonal ring structures by solving the diffraction and radiation problems. The shape of the polygonal platform or ring structure was created by using the cosine-type radial perturbation [22]. It has been shown that these hydrodynamic problems can be solved analytically by using the Eigenfunction Expansion Method. In continuing this line of study that uses the analytical method for hydrodynamic analysis of floating structures, this paper further investigates the hydrodynamic behaviour of a floating polygonal platform that is centrally placed within a floating polygonal ring structure. In order to understand the wave interactions inside the ring structure, the study will investigate the cases where the floating platform and ring structure oscillate individually as well as when they oscillate together under wave action. A parametric study involving different drafts, widths of the floating polygonal platforms and polygonal shapes will be performed to understand the hydrodynamic actions inside the floating polygonal ring structure that exhibits resonance phenomena at specific wave frequencies.

The contents of the paper are laid out as follows: Section 2 defines the problem at hand and Section 3 presents the governing equation and boundary conditions for the problem. Section 4 solves the diffracted and radiated potentials by using the semi-analytical approach. Sections 5–7 deal with the determination of the wave exciting forces, hydrodynamic radiation forces and motion responses of the floating platform and ring structure, respectively. Section 8 demonstrates the verification of the present semi-analytical approach by comparing the results with those obtained from the commercial software ANSYS AQWA. Section 9 furnishes the hydrodynamic results for parametric studies. Finally, concluding remarks are given in Section 10.

#### **2. Problem Definition**

Consider a floating rigid regular polygonal platform that is centrally placed within a floating rigid regular polygonal ring structure as shown in Figure 2. They are allowed to oscillate together or individually but are assumed to be kept in place as the current and drift force are not considered in this study. The considered water depth is *h* and the incident wave, having a period *T* and amplitude *A*, impacts the floating platform and ring structure at an oblique angle *β*. The freeboard is assumed to be sufficiently high to prevent wave overtopping. The drafts of the platform and ring structure are *d*<sup>1</sup> and *d*2, respectively.

The cylindrical coordinates (*r*, *θ*, *z*) are adopted with the origin at the centre of the regular polygonal platform.

**Figure 2.** Floating polygonal platform and ring breakwater.

The plan shape of the polygonal platform or polygonal ring structure is generated by using a radius function defined by the cosine-type radial perturbation given by [22]

$$R\_l(\theta) = R\_{0\_l} \left\{ 1 + \varepsilon\_l \cos n\_{p\_l}(\theta - \theta\_{0\_l}) \right\}, \quad l = 1, 2, 3 \tag{1}$$

where *R*0*<sup>l</sup>* , *εl*, *npl* and *θ*0*<sup>l</sup>* are parameters to be chosen by the analyst. This radius function can be used to construct all kinds of regular polygonal shapes. For example, polygonal shapes such as an equilateral triangular, square, pentagon and hexagon can be straightforwardly created by choosing the appropriate values for the dimensionless parameters *εl*, *npl* and *θ*0*l* , which are summarised in Table 1. The size and shape of the polygonal platform and ring structure are predominantly controlled by *Rl* (*l* = 1, 2, 3) where *l* = 1 represents the platform boundary whilst *l* = 2 and 3 represent the inner and outer boundaries of the ring structure, respectively. In addition, one can freely orientate the polygonal shapes by changing *θ*0*<sup>l</sup>* .

**Table 1.** Regular polygonal platform and ring shapes created from the cosine-type radial perturbation *S*0*<sup>q</sup>* denotes the plan area of the polygonal platform for *q* = 1 and the polygonal ring structure for *q* = 2. The values in the bracket are in turn associated with *R*1, *R*<sup>2</sup> and *R*3.


In this study, the following hydrodynamic properties of floating polygonal platform and ring structure are to be determined: (i) diffracted and radiated potentials, (ii) wave exciting forces, (iii) added mass, (iv) radiation damping, (v) RAOs (Response Amplitude Operators), (vi) wave field.

#### **3. Governing Equation and Boundary Conditions**

The hydrodynamic analysis will be performed in the frequency domain. The fluid domain is divided into 4 regions as shown in Figure 2. Region I is the sea space underneath the platform structure, Region II is the sea space between the platform and ring structure, Region III is the sea space underneath the ring structure and Regions IV is the sea space outside the ring structure. The fluid is assumed to be incompressible, inviscid and irrotational, and hence, the linear potential theory may be applied. Accordingly, the fluid motion is governed by the following Laplace equation

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial \phi}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2 \phi}{\partial \theta^2} + \frac{\partial^2 \phi}{\partial z^2} = 0\tag{2}$$

where *φ* is the velocity potential given by

$$
\phi = \phi\_I + \phi\_D + \phi\_R \tag{3}
$$

in which *φ<sup>I</sup>* is the incident potential, *φ<sup>D</sup>* the diffracted potential and *φ<sup>R</sup>* the radiated potential. The radiated potential may be expressed as the sum of 6 radiation modes corresponding to the six degrees of freedom as

$$\phi\_R = \sum\_{j=1}^6 \left( -i\omega \mathfrak{F}\_j \phi\_R^{(j)} \right) \tag{4}$$

where *i* is the imaginary unit, *ω* the wave angular frequency, *ξ<sup>j</sup>* the motion amplitude of the ring structure for the *<sup>j</sup>*-th radiation mode and *<sup>φ</sup>*(*j*) *<sup>R</sup>* the normalised radiated potential for the *j*-th radiation mode.

The velocity potential must satisfy the following boundary conditions

$$\left. \frac{\partial \phi}{\partial z} \right|\_{z=0} = \left. \frac{\omega^2}{\mathcal{g}} \phi \right|\_{z=0} \qquad \text{on the free surface} \tag{5}$$

$$\left.\frac{\partial\phi}{\partial z}\right|\_{z=-h} = 0 \qquad\qquad\text{on the scaled}\tag{6}$$

$$\lim\_{r \to \infty} \left( \frac{\partial \phi\_{D,R}}{\partial r} - ik \phi\_{D,R} \right) = 0 \qquad \text{at infinity} \tag{7}$$

$$
\nabla \Phi\_{\mathbf{D}} \cdot \mathbf{n}\_{\mathbf{s}\_{\parallel}} = -\nabla \Phi\_{\mathbf{I}} \cdot \mathbf{n}\_{\mathbf{s}\_{\parallel}} \qquad \text{at wetted surface for diffraction problem} \tag{8}
$$

$$\nabla \boldsymbol{\phi}\_{R}^{(j)} \cdot \mathbf{n}\_{\mathbb{6}\_l} = \mathbf{n}\_{\mathbb{J}} \cdot \mathbf{n}\_{\mathbb{6}\_l} \quad (j = 1, 2, \cdots, 6) \qquad\text{at wetted surface for radiation problem} \quad (9)$$

where *k* is the wave number, *g* the gravitational acceleration and **n***<sup>j</sup>* the generalised motion normal for 6 DOFs (degrees of freedom), i.e., **n**<sup>1</sup> = **n***x*, **n**<sup>2</sup> = **n***y*, **n**<sup>3</sup> = **n***z*, **n**<sup>4</sup> = −(*z* − *zG*)**n***<sup>y</sup>* + (*y* − *yG*)**n***z*, **n**<sup>5</sup> = (*z* − *zG*)**n***<sup>x</sup>* − (*x* − *xG*)**n***<sup>z</sup>* and **n**<sup>6</sup> = −(*y* − *yG*)**n***<sup>x</sup>* + (*x* − *xG*)**n***y*, where (*xG*, *yG*, *zG*) are the coordinates of the floating structure's centre of gravity and **n***sl* the unit normal vector to the polygonal body surface pointing out of the floating body. In order to consider the polygonal geometries, the surface function *Sl*(*r*, *θ*) = *r* − *Rl*(*θ*), (*l* = 1, 2, 3) is introduced and its derivative with respect to *θ* is defined

as *<sup>∂</sup>Sl ∂θ* <sup>=</sup> <sup>−</sup>*∂Rl*(*θ*) *∂θ* [23]. By using the surface function, the unit normal vector to the wetted body surface pointing to the fluid **n***sl* is given by [23]

$$\mathbf{n}\_{\mathbb{S}\_{l}} = \frac{\nabla S\_{l}(r,\theta)}{|\mathbf{n}\_{\mathbb{S}\_{l}}|} = \frac{\frac{\partial S\_{l} \rightarrow \stackrel{\scriptstyle \partial}{r}\mathbf{\hat{r}} + \frac{1}{r}\frac{\partial S\_{l}}{\partial \theta}\stackrel{\scriptstyle \partial}{\Theta} + \frac{\partial S\_{l}}{\partial z}\stackrel{\scriptstyle \rightarrow}{\mathbf{z}}}{\mathbf{|}|\mathbf{n}\_{\mathbb{S}\_{l}}|}} = \frac{1}{\sqrt{1 + \left(\frac{1}{r}\frac{\partial S\_{l}}{\partial \theta}\right)^{2}}} \left(\stackrel{\scriptstyle \rightarrow}{\mathbf{r}} + \frac{1}{r}\frac{\partial S\_{l}}{\partial \theta}\stackrel{\scriptstyle \rightarrow}{\Theta} + 0\stackrel{\scriptstyle \rightarrow}{\mathbf{z}}\right) \tag{10}$$

where ∇ denotes the del operator in a cylindrical coordinate system for obtaining the gradient of a vector. Thus, the normal velocity on the floating polygonal platform and ring structure wetted surface can be in general calculated by using the divergence operator in the cylindrical coordinate system as

$$\nabla \phi \cdot \mathbf{n}\_{\mathbb{G}\_l} = \frac{1}{\sqrt{1 + \left(\frac{1}{r} \frac{\partial S\_l}{\partial \theta}\right)^2}} \left(\frac{\partial \phi}{\partial r} + \frac{1}{r^2} \frac{\partial S\_l}{\partial \theta} \frac{\partial \phi}{\partial \theta}\right) \tag{11}$$

#### **4. Solutions for Diffracted and Radiated Potentials**

The assumed solutions for the diffraction and radiation problems can be expressed as a unified form as given by

$$\phi\_1^{(j,p)} = \phi\_{p\_1}^{(j,p)} + \sum\_{m=-\infty}^{\infty} \left\{ A\_{m0}^{(j,p)} \left(\frac{r}{b\_1}\right)^{|m|} + \sum\_{n=1}^{\infty} A\_{mn}^{(j,p)} \frac{I\_m(p\_n r)}{I\_m(p\_n b\_1)} \cos p\_n (z+h) \right\} \mathbf{e}^{im\theta} \tag{12}$$

$$\rho\_2^{(j,p)} = \sum\_{m=-\infty}^{\infty} \left[ \left\{ R\_{\rm m0}^{(j,p)} f\_{\rm W}(kr) + \mathcal{L}\_{\rm m0}^{(j,p)} \frac{I l\_{\rm W}(kr)}{I l\_{\rm W}(kr\_1)} \right\} \frac{Z\_{\rm Q}(z)}{Z\_{\rm Q}(l)} + \sum\_{n=1}^{\infty} \left\{ R\_{\rm m0}^{(j,p)} \frac{I\_{\rm W}(k\_{\rm W}r)}{I\_{\rm W}(k\_{\rm W}2)} + \mathcal{L}\_{\rm m0}^{(j,p)} \frac{I\_{\rm W}(k\_{\rm W}r)}{I\_{\rm W}(k\_{\rm W}a\_1)} \right\} \frac{Z\_{\rm R1}(z)}{Z\_{\rm R}(0)} \right] e^{im\theta} \tag{1.3}$$

$$\begin{aligned} \boldsymbol{\phi}\_{3}^{(j,p)} &= \boldsymbol{\phi}\_{p3}^{(j,p)} + \quad \boldsymbol{D}\_{00}^{(j,p)} \ln \frac{\boldsymbol{r}}{\boldsymbol{a}\_{2}} + \boldsymbol{E}\_{00}^{(j,p)} \ln \frac{\boldsymbol{p}\_{2}}{\boldsymbol{r}} + \quad \sum\_{m=-\infty}^{\infty} \quad \left\{ \boldsymbol{D}\_{m0}^{(j,p)} \left( \frac{\boldsymbol{r}}{\boldsymbol{a}\_{3}} \right)^{+ |m|} + \boldsymbol{E}\_{m0}^{(j,p)} \left( \frac{\boldsymbol{r}}{\boldsymbol{a}\_{2}} \right)^{-|m|} \right\} \mathbf{e}^{im\boldsymbol{\theta}} \\ &= \boldsymbol{m} + \boldsymbol{0} \\ &+ \sum\_{m=-\infty}^{\infty} \sum\_{n=1}^{\infty} \left\{ \boldsymbol{D}\_{mn}^{(j,p)} \frac{\boldsymbol{l}\_{n0}(q\_{1}r)}{\boldsymbol{l}\_{n}(q\_{1}b\_{3})} + \boldsymbol{E}\_{mn}^{(j,p)} \frac{\boldsymbol{K}\_{n0}(q\_{1}r)}{\boldsymbol{K}\_{m}^{n}(q\_{1}a\_{2})} \right\} \cos q\_{n}(\boldsymbol{z} + \boldsymbol{h}) \mathbf{e}^{im\boldsymbol{\theta}} \end{aligned} \tag{14}$$

$$\Phi\_{4}^{(j,p)} = \sum\_{m=-\infty}^{+\infty} \left\{ D\_{mn}^{(j,p)} \frac{H\_{m}^{(m,p;q-1)}}{K\_{m}(4m^{3})} + E\_{mn}^{(j,p)} \frac{K\_{m}^{(m,p;q-1)}}{K\_{m}(4n^{3}2)} \right\} \cos q\_{n}(z+h)e^{im\theta}$$

$$\Phi\_{4}^{(j,p)} = \sum\_{m=-\infty}^{\infty} \left\{ F\_{m0}^{(j,p)} \frac{H\_{m}(kr)}{H\_{m}(ka\_{3})} \frac{Z\_{0}(z)}{Z\_{0}(0)} + \sum\_{n=1}^{\infty} F\_{mn}^{(j,p)} \frac{K\_{m}(k\_{n}r)}{K\_{m}(k\_{n}a\_{3})} \frac{Z\_{n}(z)}{Z\_{n}(0)} \right\} e^{im\theta} \tag{15}$$

where the superscript *j* denotes the diffraction when *j* = 0, otherwise the radiation mode for 6 DOFs (*j* = 1 for surge, *j* = 2 for sway, *j* = 3 for heave, *j* = 4 for roll, *j* = 5 for pitch and *j* = 6 for yaw) and the superscript *p* denotes the oscillating body, i.e., *p* = 1 for the central platform, *p* = 2 for the ring structure, *p* = 3 for the two bodies oscillating monolithically or individually and *p* = 0 for zero-motion to address the diffraction problem. Hence, in order to obtain the solution for the radiation problem, the two cases for *p* = 1 and 2 must be added. *φ*(*j*,*p*) *<sup>p</sup>*<sup>1</sup> and *<sup>φ</sup>*(*j*,*p*) *<sup>p</sup>*<sup>3</sup> are respectively the particular solutions for Regions I and III and the vertical eigenfunction *Zn* for both Regions II and IV is given by

$$Z\_0(z) = \frac{\cosh k(z+h)}{\sqrt{N\_0}}, \quad N\_0 = \frac{1}{2} \left[ 1 + \frac{\sinh 2kh}{2kh} \right], (n=0) \tag{16}$$

$$Z\_n(z) = \frac{\cos k\_n(z+h)}{\sqrt{N\_n}}, \quad N\_n = \frac{1}{2} \left[ 1 + \frac{\sin 2k\_n h}{2k\_n h} \right], (n = 1, 2, \dots, \infty) \tag{17}$$

*<sup>A</sup>*(*j*,*p*) *mn* , *<sup>B</sup>*(*j*,*p*) *mn* , *<sup>C</sup>*(*j*,*p*) *mn* and *<sup>D</sup>*(*j*,*p*) *mn* are the unknown complex coefficients to be determined; *Jm* is the Bessel function of the first kind of order *m*, *Im* and *Km* are respectively the modified Bessel function of the first and the second kinds of order *m*; *al* and *bl* are respectively the shortest and the longest distance from the origin to the structure surface along the radial

direction at *r* = *Rl*(*θ*) (*l* = 1, 2, 3). The wavenumber *k*, and the vertical eigenvalue *kn* for Regions II and IV are given by

$$\frac{\omega^2}{\mathcal{S}} - k \tanh k = 0\tag{18}$$

$$k\_0 = -ik\_\prime \cdot \frac{\omega^2}{\mathcal{S}} + k\_n \tan k\_n = 0 \ (n = 1, 2, \infty) \tag{19}$$

and vertical eigenvalues *pn* and *qn* for Regions I and III are respectively given by

$$p\_n = \frac{\pi n}{h - d\_1} \ (n = 0, 1, 2, \infty) \tag{20}$$

$$q\_{\rm li} = \frac{\pi n}{\hbar - d\_2} \ (n = 0, 1, 2, \infty) \tag{21}$$

The particular solutions *<sup>φ</sup>*(*j*,*p*) *<sup>p</sup>*<sup>1</sup> and *<sup>φ</sup>*(*j*,*p*) *<sup>p</sup>*<sup>3</sup> are given by

$$\begin{split} \phi\_{p\_1}^{(j,p)}(r,\theta,z) &= -\delta\_{0j}\delta\_{0p}\phi\_I \\ &+ \frac{4(z+h)^2 - \left(1+\delta\_{ij}\right)r^2}{8(h-d\_1)} \left\{ \delta\_{3j} + \delta\_{4j} \left(r\sin\theta - y\_{C\_p}\right) - \delta\_{5j} \left(r\cos\theta - x\_{C\_p}\right) \right\} \delta\_{1p} + \frac{\left(\delta\_{4j}y\_{C\_p} - \delta\_{5j}x\_{C\_p}\right)r^2}{8(h-d\_1)} \end{split} \tag{22}$$

$$\begin{aligned} \phi\_{p\_3}^{(i,p)}(r,\theta,z) &= -\delta\_{0j}\delta\_{0p}\phi\_l\\ &+ \frac{4(z+h)^2 - (1+\delta\_{il})r^2}{8(h-d\_2)} \left\{ \delta\_{3j} + \delta\_{4j} \left(r\sin\theta - y\_{C\_p}\right) - \delta\_{5j} \left(r\cos\theta - x\_{C\_p}\right) \right\} \delta\_{2p} + \frac{(\delta\_{4j}v\_{C\_p} - \delta\_{5j}x\_{C\_p})r^2}{8(h-d\_2)} \end{aligned} \tag{23}$$

where *δij* is the Kronecker delta (1 if *i* = *j*, 0 if *i* = *j*) and (*xGp* , *yGp* ) the horizontal coordinates of the centre of gravity of the oscillating body *p*. The incident velocity potential in the cylindrical coordinate system is given by

$$\Phi\_l(r,\theta,z) = -\frac{igA}{\omega} \sum\_{m=-\infty}^{\infty} J\_m(kr) e^{im(\theta + \frac{n}{2} - \beta)} \frac{\cosh k(z+h)}{\cosh kh} \tag{24}$$

The matching conditions for the pressure and velocity continuities are

$$\nabla \Phi\_2^{(j,p)} \cdot \mathbf{n\_{51}} = \begin{cases} \{-\delta\_{0p} \nabla \phi\_I + \left(\delta\_{1p} + \delta\_{3p}\right) \mathbf{n\_{\bar{j}}}\} \cdot \mathbf{n\_{\bar{s}1}} & \text{at } r = R\_1 \text{ and } \begin{cases} -d\_1 \le z \le 0 \\ -h \le z \le -d\_1 \end{cases} \end{cases} \tag{25}$$

$$
\phi\_{1,p} = \phi\_{2,p} \quad \text{at } r = R\_1 \text{ and } -h \le z \le -d\_1 \tag{26}
$$

$$\nabla \Phi\_2^{(j,p)} \cdot \mathbf{n\_{5\_2}} = \begin{cases} \{-\delta\_{0p} \nabla \phi\_1 + \left(\delta\_{2p} + \delta\_{3p}\right) \mathbf{n\_{\bar{j}}}\} \cdot \mathbf{n\_{\bar{k}}} & \text{at } r = R\_2 \text{ and } \begin{cases} -d\_2 \le z \le 0 \\ -h \le z \le -d\_2 \end{cases} \end{cases} \tag{27}$$

$$
\phi\_{3,p} = \phi\_{2,p} \quad \text{at } r = R\_2 \text{ and } -h \le z \le -d\_2 \tag{28}
$$

$$\nabla \Phi\_4^{(j,p)} \cdot \mathbf{n\_{5\_3}} = \begin{cases} \{-\delta\_{0p} \nabla \phi\_I + \left(\delta\_{2p} + \delta\_{3p}\right) \mathbf{n\_{f}}\} \cdot \mathbf{n\_{5\_3}} & \text{at } r = R\_3 \text{ and } \begin{cases} -d\_2 \le z \le 0\\ -h \le z \le -d\_2 \end{cases} \tag{29}$$

$$
\phi\_3^{(j,p)} = \phi\_4^{(j,p)} \quad \text{at } r = R\_3 \text{ and } -h \le z \le -d\_2 \tag{30}
$$

where *Rl* (*l* = 1, 2, 3) denotes the radius function as defined in Equation (1) and **nj**·**nsl** (*l* = 1, 2, 3) for 6 DOFs is provided in Appendix A.

We consider the horizontal coordinates of the centre of gravity to coincide with the origin, i.e., *xG*<sup>1</sup> , *yG*<sup>1</sup> = *xG*<sup>2</sup> , *yG*<sup>2</sup> = (0, 0); however, the vertical coordinate of the centre of gravity may not be zero, *zG*<sup>1</sup> = 0 and *zG*<sup>2</sup> = 0. The assumed velocity potentials given in Equations (12) and (15) are substituted into the matching conditions given in Equations (25)–(30). This furnishes

$$\begin{aligned} \sum\_{\substack{nm=-\infty \ n\neq 0}}^{\infty} \sum\_{n=0}^{\infty} \left[ R\_{nm}^{(j,p)} \left\{ R\_1^2 \frac{\mathcal{T}\_{\text{min}}'(k\_1 \mathbf{R}\_1)}{\mathcal{Z}\_{\text{min}}(k\_0 \mathbf{R}\_1)} + im \frac{\mathcal{T}\_{\text{min}}(k\_1 \mathbf{R}\_1)}{\mathcal{Z}\_{\text{min}}(k\_0 \mathbf{R}\_2)} S\_{1,\theta} \right\} + \mathcal{C}\_{\text{min}}^{(j,p)} \left\{ R\_1^2 \frac{\mathcal{Z}\_{\text{max}}'(k\_1 \mathbf{R}\_1)}{\mathcal{Z}\_{\text{min}}(k\_0 \mathbf{R}\_1)} + im \frac{\mathcal{K}\_{\text{max}}(k\_0 \mathbf{R}\_1)}{\mathcal{Z}\_{\text{min}}(k\_0 \mathbf{R}\_1)} S\_{1,\theta} \right\} \right] \frac{Z\_{\text{eff}}(z)}{Z\_{\text{a}}(0)} \mathbf{e}^{im\theta} \\\ \mathbf{f} = (\delta\_{0,p} + \delta\_{1,p} + \delta\_{3,p}) \mathcal{H}\_1^{(j)} \end{aligned} \tag{31}$$
 
$$(r = R\_1, -d\_1 \le z \le 0, \ 0 \le \theta \le 2\pi)$$

∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 & *<sup>B</sup>*(*j*,*p*) *mn R*2 1 I *mn*(*knR*1) <sup>I</sup>*mn*(*knb*2) <sup>+</sup> *im*I*mn*(*knR*1) <sup>I</sup>*mn*(*knb*2) *<sup>S</sup>*1,*<sup>θ</sup>* <sup>+</sup> *<sup>C</sup>*(*j*,*p*) *mn R*2 1 K *mn*(*knR*1) <sup>K</sup>*mn*(*kna*1) *<sup>S</sup>*1,*<sup>θ</sup>* <sup>+</sup> *im*K*mn*(*knR*1) <sup>K</sup>*mn*(*kna*1) *<sup>S</sup>*1,*<sup>θ</sup>* ' *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>A</sup>*(*j*,*p*) *mn R*2 1 I *mn*(*pnR*1) <sup>I</sup>*mn*(*pnb*1) <sup>+</sup> *im*I*mn*(*pnR*1) <sup>I</sup>*mn*(*pnb*1) *<sup>S</sup>*1,*<sup>θ</sup>* cos *pn*(*z* + *h*)e*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* + *δ*1,*<sup>p</sup>* + *δ*3,*p*) ∼ P (*j*) 1 (*r* = *R*1, −*h* ≤ *z* ≤ −*d*1, 0 ≤ *θ* ≤ 2*π*) (32) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>B</sup>*(*j*,*p*) *mn* <sup>I</sup>*mn*(*knR*1) <sup>I</sup>*mn*(*knb*2) <sup>+</sup> *<sup>C</sup>*(*j*,*p*) *mn* <sup>K</sup>*mn*(*knR*1) K*mn*(*kna*1) *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>A</sup>*(*j*,*p*) *mn* <sup>I</sup>*mn*(*pnR*1) <sup>I</sup>*mn*(*pnb*1) cos *pn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)e*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*1,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*3,*p*)P(*j*) 1 (*r* = *R*1, −*h* ≤ *z* ≤ −*d*1, 0 ≤ *θ* ≤ 2*π*) (33) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 & *<sup>B</sup>*(*j*,*p*) *mn R*2 2 I *mn*(*knR*2) <sup>I</sup>*mn*(*knb*2) <sup>+</sup> *im*I*mn*(*knR*2) <sup>I</sup>*mn*(*knb*2) *<sup>S</sup>*2,*<sup>θ</sup>* <sup>+</sup> *<sup>C</sup>*(*j*,*p*) *mn R*2 2 K *mn*(*knR*2) <sup>K</sup>*mn*(*kna*1) <sup>+</sup> *im*K*mn*(*knR*2) <sup>K</sup>*mn*(*kna*1) *<sup>S</sup>*1,*<sup>θ</sup>* ' *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*2,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*3,*p*)H(*j*) 2 (*r* = *R*2, −*d*<sup>2</sup> ≤ *z* ≤ 0, 0 ≤ *θ* ≤ 2*π*) (34) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 & *<sup>B</sup>*(*j*,*p*) *mn R*2 2 I *mn*(*knR*2) <sup>I</sup>*mn*(*knb*2) <sup>+</sup> *im*I*mn*(*knR*2) <sup>I</sup>*mn*(*knb*2) *<sup>S</sup>*2,*<sup>θ</sup>* <sup>+</sup> *<sup>C</sup>*(*j*,*p*) *mn R*2 2 K *mn*(*knR*2) <sup>K</sup>*mn*(*kna*1) <sup>+</sup> *im*K*mn*(*knR*2) <sup>K</sup>*mn*(*kna*1) *<sup>S</sup>*1,*<sup>θ</sup>* ' *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 & *<sup>D</sup>*(*j*,*p*) *mn R*2 2 I *mn*(*qnR*2) <sup>I</sup>*mn*(*qnb*3) <sup>+</sup> *im*I*mn*(*qnR*2) <sup>I</sup>*mn*(*qnb*3) *<sup>S</sup>*2,*<sup>θ</sup>* <sup>+</sup>*E*(*j*,*p*) *mn R*2 2 K *mn*(*qnR*2) <sup>K</sup>*mn*(*qna*2) <sup>+</sup> *im*K*mn*(*qnR*2) <sup>K</sup>*mn*(*qna*2) *<sup>S</sup>*2,*<sup>θ</sup>* ' <sup>+</sup> cos *qn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)e*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*2,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*3,*p*) ∼ Q (*j*) 2 (*r* = *R*2, −*h* ≤ *z* ≤ −*d*2, 0 ≤ *θ* ≤ 2*π*) (35) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>B</sup>*(*j*,*p*) *mn* <sup>I</sup>*mn*(*knR*2) <sup>I</sup>*mn*(*knb*2) <sup>+</sup> *<sup>C</sup>*(*j*,*p*) *mn* <sup>K</sup>*mn*(*knR*2) K*mn*(*kna*1) *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>D</sup>*(*j*,*p*) *mn* <sup>I</sup>*mn*(*qnR*2) <sup>I</sup>*mn*(*qnb*3) <sup>+</sup> *<sup>E</sup>*(*j*,*p*) *mn* <sup>K</sup>*mn*(*qnR*2) K*mn*(*qna*2) cos *qn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)e*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*2,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*3,*p*)Q(*j*) 2 (*r* = *R*2, −*h* ≤ *z* ≤ −*d*2, 0 ≤ *θ* ≤ 2*π*) (36) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>F</sup>*(*j*,*p*) *mn R*2 3 K *mn*(*knR*3) <sup>K</sup>*mn*(*kna*3) <sup>+</sup> *im*K*mn*(*knR*3) <sup>K</sup>*mn*(*kna*3) *<sup>S</sup>*3,*<sup>θ</sup> Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>=</sup> *δ*0,*<sup>p</sup>* + *δ*2,*<sup>p</sup>* + *δ*3,*<sup>p</sup>* H(*j*) 3 (*r* = *R*3, −*d*<sup>2</sup> ≤ *z* ≤ 0, 0 ≤ *θ* ≤ 2*π*) (37) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>F</sup>*(*j*,*p*) *mn R*2 3 K *mn*(*knR*3) <sup>K</sup>*mn*(*kna*3) <sup>+</sup> *im*K*mn*(*knR*3) <sup>K</sup>*mn*(*kna*3) *<sup>S</sup>*3,*<sup>θ</sup> Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 & *<sup>D</sup>*(*j*,*p*) *mn R*2 3 I *mn*(*qnR*3) <sup>I</sup>*mn*(*qnb*3) <sup>+</sup> *im*I*mn*(*qnR*3) <sup>I</sup>*mn*(*qnb*3) *<sup>S</sup>*3,*<sup>θ</sup>* <sup>+</sup>*E*(*j*,*p*) *mn R*2 3 K *mn*(*qnR*3) <sup>K</sup>*mn*(*qna*2) <sup>+</sup> *im*K*mn*(*qnR*3) <sup>K</sup>*mn*(*qna*2) *<sup>S</sup>*3,*<sup>θ</sup>* ' cos *qn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)e*im<sup>θ</sup>* = (*δ*0,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*2,*<sup>p</sup>* <sup>+</sup> *<sup>δ</sup>*3,*p*) ∼ Q (*j*) 3 (*r* = *R*3, −*h* ≤ *z* ≤ −*d*2, 0 ≤ *θ* ≤ 2*π*) (38) ∞ <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>F</sup>*(*j*,*p*) *mn* <sup>K</sup>*mn*(*knR*3) K*mn*(*kna*3) *Zn*(*z*) *Zn*(0) <sup>e</sup>*im<sup>θ</sup>* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 *<sup>D</sup>*(*j*,*p*) *mn* <sup>I</sup>*mn*(*qnR*3) <sup>I</sup>*mn*(*qnb*3) <sup>+</sup> *<sup>E</sup>*(*j*,*p*) *mn* <sup>K</sup>*mn*(*qnR*3) K*mn*(*qna*2) cos *qn*(*z* + *h*)e*im<sup>θ</sup>* = *δ*0,*<sup>p</sup>* + *δ*2,*<sup>p</sup>* + *δ*3,*<sup>p</sup>* Q(*j*) 3 (*r* = *R*3, −*h* ≤ *z* ≤ −*d*2, 0 ≤ *θ* ≤ 2*π*) (39)

where *Sl*,*<sup>θ</sup>* denotes *<sup>∂</sup>Sl ∂θ* ( ( ( *r*=*Rl*(*θ*) (*l* = 1, 2, 3), K *mn* and I *mn* are respectively the derivatives of K*mn* and I*mn* with respect to *r*. I*mn*(*knr*), I*mn*(*pnr*), I*mn*(*qnr*), K*mn*(*qnr*) and K*mn*(*knr*) are given by

$$\mathcal{L}\_{mn}(k\_nr) = \begin{cases} f\_m(kr) & \text{for } n = 0\\ \frac{I\_m(k\_nr)}{I\_m(k\_nb\_2)} & \text{for } n > 0 \end{cases} \tag{40}$$

$$\mathcal{I}\_{mn}(p\_n r) = \begin{cases} \left(\frac{r}{b\_1}\right)^{|m|} & \text{for } n = 0\\ \frac{l\_m(p\_n r)}{l\_m(p\_n b\_1)} & \text{for } n > 0 \end{cases} \tag{41}$$

$$\mathcal{I}\_{mn}(q\_nr) = \begin{cases} \ln \frac{r}{a\_2} & \text{for } n = 0 \text{ and } m = 0\\ \left(\frac{r}{b\_3}\right)^{|m|} & \text{for } n = 0 \text{ and } m \neq 0\\ \frac{I\_m(q\_nr)}{I\_m(q\_nb\_2)} & \text{for } n > 0 \end{cases} \tag{42}$$

$$\mathcal{K}\_{mn}(q\_nr) = \begin{cases} \ln \frac{b\_3}{r} & \text{for } n = 0 \text{ and } m = 0\\ \left(\frac{r}{a\_2}\right)^{-|m|} & \text{for } n = 0 \text{ and } m \neq 0\\ \frac{K\_m(q\_nr)}{K\_m(q\_nb\_2)} & \text{for } n > 0 \end{cases} \tag{43}$$

$$\mathcal{K}\_{mn}(k\_nr) = \begin{cases} \frac{H\_m(kr)}{H\_m(ka\_3)} & \text{for } n = 0\\ \frac{K\_m(k\_nr)}{K\_m(k\_na\_3)} & \text{for } n > 0 \end{cases} \tag{44}$$

H(*j*) *<sup>l</sup>* <sup>P</sup>(*j*) *<sup>l</sup>* , <sup>Q</sup>(*j*) *l* , ∼ P (*j*) *<sup>l</sup>* and <sup>∼</sup> Q (*j*) *<sup>l</sup>* (*j* = 0, 1, ··· , 6; *l* = 1, 2, 3) are provided in Appendix B. In the foregoing formulation, numerical integration should be used for establishing the associated simultaneous equations because the radius functions are substituted into the arguments of radial terms such as the Bessel function, modified Bessel function and radial polynomial functions. Liu, et al. [24] replaced such radial terms with the Fourier expansion series so that the integration could be performed analytically. Recently, Park and Wang [20] suggested an improved method that enables the computational time to be significantly reduced by using minimal numbers of Fourier coefficient sets. By using Park and Wang [20] improved method, 13 sets of Fourier coefficients such as *a* (1) *<sup>m</sup>*,*n*,*q*, *b* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , *c* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , *<sup>d</sup>* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , *<sup>e</sup>* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , *<sup>f</sup>* (3) *<sup>m</sup>*,*<sup>q</sup>* and *<sup>g</sup>* (1,2,3) *<sup>m</sup>*,*<sup>q</sup>* are introduced for solving diffraction and radiation problems. The applied Fourier expansions are as given in Table 2 and their derivatives with respect to *r* can be directly obtained by using the formerly obtained Fourier coefficients (see Table 3). The Fourier coefficients can be calculated by the functional orthogonality, i.e., by multiplying *e*−*iq<sup>θ</sup>* and integrating over [0, 2*π*].

**Table 2.** Fourier expansions for the functions used in velocity potential.



**Table 3.** Fourier expansions for the derivative of functions used in velocity potential.

Also, the radius function and the derivative of the surface function with respect to *θ* can be obtained by using *a*1,0,*q*, *d*1,0,*<sup>q</sup>* and *e*1,0,*q*.

$$|r|\_{r=R\_1(\theta)} = \sum\_{n\_r=-\infty}^{\infty} a\_{1,0,n\_r}^{(1)} \mathbf{e}^{in\_r\theta} \tag{45}$$

$$\left.r\right|\_{r=R\_{2,3}(\theta)} = \sum\_{n\_r=-\infty}^{\infty} d\_{1,0,n\_r}^{(2,3)} \mathbf{e}^{in\_r\theta} \tag{46}$$

$$\left. \frac{1}{r} \right|\_{r=R\_{2,3}(\theta)} = \sum\_{n\_r=-\infty}^{\infty} e\_{1,0,n\_r}^{(2,3)} e^{in\_r \theta} \tag{47}$$

$$\left. \frac{\partial \mathcal{S}\_1}{\partial \theta} \right|\_{S\_1 = 0} = - \sum\_{n\_r = -\infty}^{\infty} (i n\_r) a\_{1, 0, n\_r}^{(1)} \mathbf{e}^{i n\_r \theta} \tag{48}$$

$$\left. \frac{\partial S\_{2,3}}{\partial \theta} \right|\_{S\_{2,3}=0} = -\sum\_{n\_r=-\infty}^{\infty} (in\_r) d\_{1,0,n\_r}^{(2,3)} \mathbf{e}^{in\_r \theta} \tag{49}$$

Equations (31)–(39) can be thus expressed by using the Fourier coefficients obtained in Tables 2 and 3 as

$$\sum\_{m=-\infty}^{\infty} \sum\_{n\_l=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_l+q)\theta} \sum\_{n=0}^{\infty} \left[ B\_{mn}^{(j,p)} \hat{b}\_{m,n,q}^{'} + \mathbf{C}\_{mn}^{(j,p)} \hat{\mathbf{c}}\_{m,n,q}^{'} \right] \frac{Z\_0(z)}{Z\_0(0)} = \left( \delta\_{0,p} + \delta\_{1,p} + \delta\_{3,p} \right) \mathcal{H}\_1^{(j)} \tag{50}$$
 
$$\left( r = \tilde{R}\_1, -d\_1 \le z \le 0, \ 0 \le \theta \le 2\pi \right)$$

$$\begin{aligned} \sum\_{\substack{m=-\infty \ n,\,\tau=-\infty}}^{\infty} \sum\_{n=-\infty}^{\infty} \underbrace{\mathcal{C}^{j(m+n+q)\theta}}\_{=-\infty} \otimes \sum\_{n=0}^{\infty} \left[ B\_{mn}^{(j,p)} \overset{\leftrightarrow}{b}\_{m,n,q}^{(1)} + \mathcal{C}\_{mn}^{(j,p)} \overset{\leftrightarrow}{c}\_{m,n,q}^{(1)} \right] \frac{Z\_n(z)}{\mathcal{Z}\_0(\theta)} &= \sum\_{m=-\infty}^{\infty} \sum\_{n=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathcal{C}^{j(m+n+q)\theta} \sum\_{n=0}^{\infty} A\_{mn}^{(j,p)} \overset{\leftrightarrow}{a}\_{m,n,q}^{(1)} \\ &- (\delta\_{0,p} + \delta\_{1,p} + \delta\_{3,p}) \overset{\leftrightarrow}{\mathcal{P}}\_{1}^{(j)} \end{aligned} \tag{51}$$
 
$$\left( r - R\_{1,r} - h \overset{\leftrightarrow}{\le} z \right) \stackrel{\leftrightarrow}{\mathcal{P}}\_{1}^{(j)} \left( r - R\_{1,r} - h \overset{\leftrightarrow}{\le} z \right) - d\_1, \ 0 \le \theta \le 2\pi \right)$$

$$\begin{cases} \sum\_{\substack{m=-\infty \ q=-\infty}}^{\infty} \sum\_{q=-\infty}^{\infty} q^{j(m+q)\theta} \sum\_{n=0}^{\infty} \left\{ B\_{mn}^{(\rho,\rho)} \overset{\rho^{+}}{b}\_{m,n,q} + \mathcal{L}\_{mn}^{(\boldsymbol{\beta},\boldsymbol{\rho})} \overset{\boldsymbol{\varsigma}}{\mathcal{L}\_{m,n,q}} \right\} \frac{\mathcal{Z}\_{\boldsymbol{\varsigma}}(z)}{\mathcal{Z}\_{\boldsymbol{\varsigma}}(\boldsymbol{0})} - \sum\_{m=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} q^{j(m+q)\theta} \sum\_{n=0}^{\infty} A\_{mn}^{(\boldsymbol{\beta},\boldsymbol{\rho}) - \boldsymbol{\varsigma}^{\prime}} \cos p\_{n}(z+h) \\\ - (\delta\_{0,p} + \delta\_{1,p} + \delta\_{3,p}) \mathcal{P}\_{1}^{(\boldsymbol{\beta})} \end{cases} \tag{52}$$
 
$$\left\{ r - R\_{1r} - h \le z \le -d\_{1r} \, 0 \le \theta \le 2\pi \right\}$$

$$\sum\_{m=-\infty}^{\infty} \sum\_{n\_l=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_l+q)\theta} \sum\_{n=0}^{\infty} \left[ B\_{mn}^{(j,p)} \hat{b}\_{m,n,q}^{\gamma'} + \mathbf{C}\_{mn}^{(j,p)} \hat{c}\_{m,n,q}^{\gamma'} \right] \frac{Z\_n(z)}{Z\_0(0)} = \left( \delta\_{0,p} + \delta\_{2,p} + \delta\_{3,p} \right) \mathcal{H}\_2^{(j)} \tag{53}$$
 
$$(r = \bar{R}\_2, -d\_2 \le z \le 0, \ 0 \le \theta \le 2\pi)$$

$$\begin{aligned} \sum\_{m=-\infty}^{\infty} \sum\_{n\_l=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_l+q)\theta} \sum\_{n=0}^{\infty} \left[ B\_{mn}^{(j,p)} \overset{\sim}{b}\_{m,n,q}^{\prime} + \underset{\sigma \to \infty}{\operatorname{em}} \overset{\sim}{\operatorname{\acute{e}}\_{m,n,q}^{\prime}} \right] \frac{Z\_{n\ell}(z)}{Z\_{\ell}(0)} \\ - \sum\_{m=-\infty}^{\infty} \sum\_{n\_l=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_l+q)\theta} \sum\_{n=0}^{\infty} \left[ D\_{mn}^{(j,p)} \overset{\sim}{d}\_{m,n,q}^{\prime} + \underset{\sigma \to \infty}{\operatorname{em}} \overset{\sim}{\operatorname{\acute{e}}\_{m,n,q}^{\prime}} \right] \cos q\_{n}(z+h) \\ = \left( \delta\_{0,p} + \delta\_{2,p} + \delta\_{3,p} \right) \overset{\sim}{Q}\_{2} \end{aligned} \tag{54}$$
 
$$(r = R\_{2\nu} - h \le z \le -d\_{2\nu}, 0 \le \theta \le 2\pi)$$

$$\begin{aligned} \sum\_{\substack{m=-\infty \\ m=-\infty}}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+q)\theta} \sum\_{n=0}^{\infty} \left[ B\_{mn}^{(j,p)} \hat{b}^{(2)}\_{m,n;q} + \hat{c}\_{mn}^{(j,p)} \hat{c}\_{m,n;q}^{(2)} \right] \frac{Z\_{a}(z)}{Z\_{a}(0)} \\ - \sum\_{m=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+q)\theta} \sum\_{n=0}^{\infty} \left[ D\_{mn}^{(j,p)} \hat{d}\_{m,n;q}^{(2)} + E\_{mn}^{(j,p)} \hat{c}\_{m,n;q}^{(2)} \right] \cos q\_{n}(z+h) &= \left( \delta\_{0,p} + \delta\_{2,p} + \delta\_{3,p} \right) Q\_{2}^{(j)} \\ (r = R\_{2}, -h \le z \le -d\_{2}, \ 0 \le \theta \le 2n) \end{aligned} \tag{55}$$

$$\sum\_{n=-\infty}^{\infty} \sum\_{n\_r=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_r+q)\theta} \sum\_{n=0}^{\infty} F\_{mn}^{(j,p)} \hat{\boldsymbol{f}}\_{m,n,q}^{(3)} \frac{Z\_n(z)}{Z\_n(0)} = \left(\delta \mathbf{0}\_{\mathcal{P}} + \delta \mathbf{1}\_{\mathcal{P}} + \delta \mathbf{3}\_{\mathcal{P}}\right) \mathcal{H}\_3^{(j)}\tag{56}$$
 
$$\left(r = R\_3, -d\_2 \le z \le 0, \ 0 \le \theta \le 2\pi\right)$$

$$\begin{aligned} &\sum\_{m=-\infty}^{\infty} \sum\_{n\_r=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_r+q)\theta} \sum\_{n=0}^{\infty} F\_{mn}^{(j,p)} \widetilde{\boldsymbol{f}}\_{m,n,q}^{(3)} \frac{\boldsymbol{Z}\_m(z)}{\boldsymbol{Z}\_n(0)} \\ &- \sum\_{m=-\infty}^{\infty} \sum\_{n\_r=-\infty}^{\infty} \sum\_{q=-\infty}^{\infty} \mathbf{e}^{i(m+n\_r+q)\theta} \sum\_{n=0}^{\infty} \left[ D\_{mn}^{(j,p)} \widetilde{\boldsymbol{d}}\_{m,n,q}^{(3)} + E\_{mn}^{(j,p)} \widetilde{\boldsymbol{e}}\_{m,n,q}^{(3)} \right] \cos q\_n(z+h) \\ &= (\delta\_{0,p} + \delta\_{2,p} + \delta\_{3,p}) \widetilde{Q}\_3 \end{aligned} \tag{57}$$

$$\begin{aligned} \left(r = R\_3, -h \le z \le -d\_2, \; 0 \le \theta \le 2\pi\right) \\ \sum\_{m = -\infty}^{\infty} \sum\_{q = -\infty}^{\infty} q^{((m+q)\theta + \frac{\pi}{2})} \sum\_{n=0}^{\infty} \int\_{m,n,q}^{(j,p)} \overline{f}\_{n,n,q}^{(2)} \frac{\mathbb{Z}\_n(z)}{\mathbb{Z}\_n(0)} - \sum\_{m = -\infty}^{\infty} \sum\_{q = -\infty}^{\infty} q^{((m+q)\theta + \frac{\pi}{2})} \sum\_{n=0}^{\infty} \left[ D\_{mn}^{(j,p)} \overline{d}\_{m,n,q}^{(3)} + \overline{\nu}\_{mn}^{(j)} \overline{\nu}\_{n,n,q}^{(3)} \right] \cos q\_n (z + h) \\ - \left(\delta\_{0,p} + \delta\_{2,p} + \delta\_{3,p}\right) Q\_3^{(j)} \end{aligned} \tag{58}$$

where the reduced forms of <sup>∼</sup> *a* (1) *<sup>m</sup>*,*n*,*q*, ∼ *b* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *c* (1),(2) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *d* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *e* (2),(3) *<sup>m</sup>*,*n*,*<sup>q</sup>* , ∼ *f* (3) *<sup>m</sup>*,*n*,*<sup>q</sup>* and their normal derivatives are given in Appendix C.

By multiplying the corresponding vertical eigenfunctions <sup>1</sup> *<sup>h</sup>Zn*(*z*) in Equations (50), (51), (53), (54), (56) and (57), <sup>1</sup> *<sup>h</sup>*−*d*<sup>1</sup> cos *pn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*) in Equation (52), <sup>1</sup> *<sup>h</sup>*−*d*<sup>2</sup> cos *qn*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*) in Equations (55) and (58) and the angular eigenfunction *e*−*im<sup>θ</sup>* in Equations (50)–(58) and integrating the equations for the associated integral intervals, one can combine Equation (50) with (51), Equation (53) with (54) and Equation (56) with (57). By truncating the series terms at *m* = *M*, *n* = *N*, *nr* = *Nr* and *q* = *Nq*, 9(2*M* + 1)(*N* + 1) equations for the monolithic motion or 18(2*M* + 1)(*N* + 1) equations for the individual motion, and the same number

of unknowns are given for the diffraction problem and each radiation mode. Consequently, the unknown complex coefficients (*A*(*j*,*p*) *mn* , *<sup>B</sup>*(*j*,*p*) *mn* , *<sup>C</sup>*(*j*,*p*) *mn* , *<sup>D</sup>*(*j*,*p*) *mn* , *<sup>E</sup>*(*j*,*p*) *mn* , *<sup>F</sup>*(*j*,*p*) *mn* ) can be solved by linear algebra.

#### **5. Determination of Wave Exciting Force**

From Bernoulli's equation, the fluid pressure *p* is given by

$$
\varphi = \rho i \omega \phi \tag{59}
$$

where *ρ* is the water density.

By integrating over the wetted areas, the wave exciting force and rotational moment for 6 DOFs are obtained by

$$F\_{w\_j}^{(q)} = \rho i\omega \int\_{S\_{w\_q}} (\phi\_I + \phi\_D) \cdot \mathbf{n}\_{\dot{f}} \mathrm{d}S\_{w\_q} \tag{60}$$

where *Swq* denotes the wetted surface of the floating body *q* on which the wave exciting forces are acting.

By substituting the incident velocity potential and diffracted potentials with obtained unknown complex coefficients into Equation (60), the wave exciting forces in 3 DOFs (i.e., surge, heave and pitch) acting on the floating polygonal platform (*q* = 1) and ring structure (*q* = 2) are given below:

*<sup>F</sup>*(1) *<sup>w</sup>*<sup>1</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>1</sup> *φ<sup>I</sup>* + *φD*<sup>2</sup> ( ( *r*=*R*<sup>1</sup> *R*1**n***x*·(−**ns**<sup>1</sup> )d*z*d*θ* = *<sup>ρ</sup>gA* cosh *kh* ∞ <sup>∑</sup>*m*=−<sup>∞</sup> e*im*( *<sup>π</sup>* <sup>2</sup> <sup>−</sup>*β*) \* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> *Jm*(*kR*1)e*imθR*1**n**1·(−**ns**<sup>1</sup> ) <sup>d</sup>*<sup>θ</sup>* \* 0 <sup>−</sup>*d*<sup>1</sup> cosh *<sup>k</sup>*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)d*<sup>z</sup>* <sup>+</sup>*ρi<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> {*Bmn*I*mn*(*knR*1) <sup>+</sup> *Cmn*K*mn*(*knR*2)}e*imθR*1**n**1·(−**n***s*<sup>1</sup> )d*<sup>θ</sup>* \* 0 −*d*<sup>1</sup> *Zn*(*z*) *Zn*(0)d*z* (61) *<sup>F</sup>*(1) *<sup>w</sup>*<sup>3</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>2</sup> *R*1 *φ<sup>I</sup>* + *φD*<sup>2</sup> ( ( *z*=−*d*<sup>1</sup> **nz**·(**nz**)*r*d*r*d*θ* <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ cos *pn*(*h* − *d*1) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>1</sup> <sup>0</sup> *Amn*I*mn*(*pnr*)e*im<sup>θ</sup> <sup>r</sup>*d*r*d*<sup>θ</sup>* (62)

*<sup>F</sup>*(1) *<sup>w</sup>*<sup>5</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 &\* <sup>0</sup> −*d*<sup>1</sup> *z* − *zG*<sup>1</sup> *φ<sup>I</sup>* <sup>+</sup> *<sup>φ</sup>D*<sup>2</sup> ( ( *r*=*R*<sup>1</sup> *<sup>R</sup>*1**n***x*·(−**ns**<sup>1</sup> )d*<sup>z</sup>* <sup>−</sup> \* *<sup>R</sup>*<sup>1</sup> 0 *φ<sup>I</sup>* + *φD*<sup>1</sup> ( ( *z*=−*d*<sup>1</sup> (*r* cos *θ*)**nz**·(**n***z*)*r*d*r* ' d*θ* = *<sup>ρ</sup>gA* cosh *kh* ∞ <sup>∑</sup>*m*=−<sup>∞</sup> e*im*( *<sup>π</sup>* <sup>2</sup> −*β*) &\* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> *Jm*(*kR*1)*R*1**n**1·(−**n***s*<sup>1</sup> )e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>1</sup> *z* − *zG*<sup>1</sup> cosh *k*(*z* + *h*)d*z* − cosh *k*(*h* − *d*1) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>1</sup> <sup>0</sup> *Jm*(*kr*)e*im<sup>θ</sup> <sup>r</sup>*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* ' <sup>+</sup>*ρi<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 &\* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> {*Bmn*I*mn*(*knR*1) <sup>+</sup> *Cmn*K*mn*(*knR*1)}e*imθR*1**n**1·(−**n***s*<sup>1</sup> )d*<sup>θ</sup>* \* 0 −*d*<sup>1</sup> *z* − *zG*<sup>1</sup> *Zn*(*z*) *Zn*(0)d*z* −cos *pn*(*h* − *d*1) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>1</sup> <sup>0</sup> *Amn*I*mn*(*pnr*)e*im<sup>θ</sup> <sup>r</sup>*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* ' (63)

*<sup>F</sup>*(2) *<sup>w</sup>*<sup>1</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>2</sup> *φ<sup>I</sup>* + *φD*<sup>2</sup> ( ( *r*=*R*<sup>2</sup> *<sup>R</sup>*2**nx**·**ns**<sup>2</sup> <sup>+</sup> *φ<sup>I</sup>* + *φD*<sup>4</sup> ( ( *r*=*R*<sup>3</sup> *R*3**nx**·(−**ns**<sup>3</sup> ) d*z*d*θ* = *<sup>ρ</sup>gA* cosh *kh* ∞ <sup>∑</sup>*m*=−<sup>∞</sup> e*im*( *<sup>π</sup>* <sup>2</sup> <sup>−</sup>*β*) \* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> {*Jm*(*kR*2)*R*2**n**1·**ns**<sup>2</sup> <sup>+</sup> *Jm*(*kR*3)*R*3**n**1·(−**ns**<sup>3</sup> )}e*im<sup>θ</sup>* <sup>d</sup>*<sup>θ</sup>* \* 0 <sup>−</sup>*d*<sup>2</sup> cosh *<sup>k</sup>*(*<sup>z</sup>* <sup>+</sup> *<sup>h</sup>*)d*<sup>z</sup>* <sup>+</sup>*ρi<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> [{*Bmn*I*mn*(*knR*2) <sup>+</sup> *CmnKmn*(*knR*2)}*R*2**n**1·**n***s*<sup>2</sup> <sup>+</sup> *Fmn*K*mn*(*knR*3)*R*3**n**<sup>1</sup> ·(−**n***s*<sup>3</sup> )]e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> *Zn*(*z*) *Zn*(0)d*z* (64) *<sup>F</sup>*(2) *<sup>w</sup>*<sup>3</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 *φ<sup>I</sup>* + *φD*<sup>3</sup> ( ( *z*=−*d*<sup>2</sup> **nz**·(**nz**)*r*d*r*d*θ* <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ cos *pn*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> {*Dmn*I*mn*(*qnr*) <sup>+</sup> *Emn*K*mn*(*qnr*)}e*im<sup>θ</sup> <sup>r</sup>*d*r*d*<sup>θ</sup>* (65)

*n*=0

*n*=0

*<sup>F</sup>*(2) *<sup>w</sup>*<sup>5</sup> <sup>=</sup> *<sup>ρ</sup>i<sup>ω</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 &\* <sup>0</sup> −*d*<sup>2</sup> *z* − *zG*<sup>2</sup> *φ<sup>I</sup>* + *φD*<sup>2</sup> ( ( *r*=*R*<sup>2</sup> *<sup>R</sup>*2**n***x*·**ns**<sup>2</sup> <sup>+</sup> *φ<sup>I</sup>* + *φD*<sup>4</sup> ( ( *r*=*R*<sup>3</sup> *R*3**n***x*·(−**ns**<sup>3</sup> ) d*z* <sup>−</sup> \* *<sup>R</sup>*<sup>3</sup> *R*2 *φ<sup>I</sup>* + *φD*<sup>3</sup> ( ( *z*=−*d*<sup>2</sup> (*r* cos *θ*)**nz**·(**n***z*)*r*d*r* ' d*θ* = *<sup>ρ</sup>gA* cosh *kh* ∞ <sup>∑</sup>*m*=−<sup>∞</sup> e*im*( *<sup>π</sup>* <sup>2</sup> −*β*) &\* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> {*Jm*(*kR*2)*R*2**n**1·**n***s*<sup>2</sup> <sup>+</sup> *Jm*(*kR*3)*R*3**n**1·**n***s*<sup>3</sup> }e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> (*z* −*zG*<sup>2</sup> cosh *k*(*z* + *h*)d*z* − cosh *k*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> *Jm*(*kr*)e*im<sup>θ</sup> <sup>r</sup>*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* ' <sup>+</sup>*ρi<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* <sup>0</sup> [{*Bmn*I(*knR*2) + *Cmn*K*mn*(*knR*2)}*R*2**n**1·**n***s*<sup>2</sup> + *Fmn*K*mn*(*knR*3)*R*3**n**<sup>1</sup> ·(−**n***s*<sup>3</sup> )]e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> *z* − *zG*<sup>2</sup> *Zn*(*z*) *Zn*(0)d*z* <sup>−</sup>*ρi<sup>ω</sup>* <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 cos *pn*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> {*Dmn*I*mn*(*qnr*) <sup>+</sup> *Emn*K*mn*(*qnr*)}e*im<sup>θ</sup> <sup>r</sup>*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* (66)

where the unknown complex coefficients for the diffraction *<sup>A</sup>*(0,0) *mn* , *<sup>B</sup>*(0,0) *mn* , *<sup>C</sup>*(0,0) *mn* , *<sup>D</sup>*(0,0) *mn* , *<sup>E</sup>*(0,0) *mn* and *<sup>F</sup>*(0,0) *mn* are expressed as *Amn*, *Bmn*, *Cmn*, *Dmn*, *Emn* and *Fmn* for brevity. *zG*<sup>1</sup> and *zG*<sup>2</sup> are the z-coordinates of the centre of gravity for the floating polygonal platform and ring structure, respectively. The wave exciting forces for other DOFs such as sway, roll and yaw can be similarly obtained by using the unit normal vector (−**ns** + **n***z*) and the generalised motion normal **n***j*. The divergence **n***j*·**n***<sup>s</sup>* are given in Appendix A.

#### **6. Determination of Radiation Forces**

The radiation force is obtained by

$$F\_{\rm ij}^{(q,p)} = i\omega\rho \int\_{S\_{w\_q}} \left\{-i\omega \xi\_j^{(p)} \phi\_{\mathbb{R}}^{(j,p)}\right\} \mathbf{n}\_j \mathbf{dS}\_{w\_q} = \xi\_j \left(\omega^2 \mu\_{\rm ij}^{(q,p)} + i\omega \lambda\_{\rm ij}^{(q,p)}\right) \tag{67}$$

where *<sup>μ</sup>*(*q*,*p*) *ij* and *<sup>λ</sup>*(*q*,*p*) *ij* are the added mass and the radiation damping of the floating body *q* by the oscillating body *p* for the *i*-th mode of force and the *j*-th mode of motion, which are respectively defined as

$$
\mu\_{ij}^{(q,p)} = \text{Re}\left(\rho f\_{ij}^{(q,p)}\right) \tag{68}
$$

$$
\lambda\_{ij}^{(q,p)} = \text{Im}\left(\rho \omega f\_{ij}^{(q,p)}\right) \tag{69}
$$

Re(·) denotes the real part and Im(·) the imaginary part. *f* (*q*,*p*) *ij* is the integral form for *i*-th mode of force and *j*-th mode of motion associated with the floating body *q* by the oscillating body *p*. By substituting the normalised radiated velocity potentials with the obtained unknown complex coefficients into Equation (67), *f* (*q*,*p*) *ij* (*p* = 1, 2 and *q* = 1, 2) for 3 DOFs are given by

$$\begin{split} f\_{11}^{(1,p)} &= \int\_{0}^{2\pi} \int\_{-d\_{1}}^{0} \phi\_{R\_{2}}^{(1,p)} \Big|\_{r=R\_{1}} \mathbf{n}\_{4} \cdot (-\mathbf{n}\_{01}) \mathcal{R}\_{1} \mathrm{d}z \mathrm{d}\theta \\ &= \sum\_{m=-\infty}^{\infty} \sum\_{n=0}^{\infty} \int\_{0}^{2\pi} \left\{ \left\{ B\_{mn}^{(1,p)} \mathcal{Z}\_{mn}(k\_{n} \mathcal{R}\_{1}) + \mathcal{C}\_{mn}^{(1,p)} \mathcal{K}\_{mn}(k\_{n} \mathcal{R}\_{1}) \right\} \mathcal{R}\_{1} \mathbf{n}\_{4} \cdot (-\mathbf{n}\_{01}) \right\} \mathrm{e}^{im\theta} \mathrm{d}\theta \int\_{-d\_{1}}^{0} \frac{\mathcal{Z}\_{n}(z)}{Z\_{n}(0)} \mathrm{d}z \end{split} \tag{70}$$

$$\begin{split} &f\_{51}^{(1,p)} = \int\_{0}^{2\pi} \Big\{\int\_{-d\_{1}}^{0} (z - z\_{G\_{1}}) \phi\_{R\_{2}}^{(1,p)} \Big|\_{r = R\_{1}} \mathbf{n}\_{\mathbf{x}} \cdot (-\mathbf{n}\_{\mathbf{u}}) R\_{1} \mathrm{d}z - \int\_{0}^{R\_{1}} \phi\_{R\_{1}}^{(1,p)} \Big|\_{z = -d\_{1}} (r \cos \theta) \mathbf{n}\_{\mathbf{x}} \cdot (\mathbf{n}\_{\mathbf{z}}) r \mathrm{d}r \Big\} \mathrm{d}\theta \\ &= \sum\_{m = -\infty}^{\infty} \sum\_{n=0}^{\infty} \int\_{0}^{2\pi} \Big\{\mathcal{B}\_{mn}^{(1,p)} \mathcal{Z}\_{mn} (k\_{n} R\_{1}) + \mathcal{C}\_{nm}^{(1,p)} \mathcal{K}\_{nm} (k\_{n} R\_{1}) \right\} R\_{1} \mathbf{n}\_{\mathbf{x}} \cdot (-\mathbf{n}\_{\mathbf{u}}) \mathrm{e}^{im\theta} \mathrm{d}\theta \int\_{-d\_{1}}^{0} (z - z\_{G\_{1}}) \frac{\mathcal{Z}\_{n} (z)}{\mathcal{Z}\_{n} (0)} \mathrm{d}z \\ &- \sum\_{m = -\infty}^{\infty} \sum\_{n=0}^{\infty} \cos p\_{n} (h - d\_{1}) \int\_{0}^{2\pi} \int\_{0}^{R\_{1}} A\_{nm}^{(1,p)} \mathcal{Z}\_{mn} (p\_{n} r) \mathrm{e}^{im\theta} r^{2} \cos \theta \mathrm{d}r \mathrm{d}\theta \\ &\qquad \qquad \qquad \qquad (11.10) \end{split}$$

$$\begin{split} f\_{33}^{(1,p)} &= \int\_{0}^{2\pi} \int\_{0}^{R\_{1}} \Phi\_{R\_{2}}^{(3,p)} \Big|\_{z=-d\_{1}} \mathbf{n}\_{\mathbf{z}} \cdot (\mathbf{n}\_{z}) r \mathrm{d}r \mathrm{d}\theta \\ &= \delta\_{1p} \int\_{0}^{2\pi} \int\_{0}^{R\_{1}} \left\{ \frac{(h-d\_{1})}{2} - \frac{r^{2}}{4(h-d\_{1})} \right\} r \mathrm{d}r \mathrm{d}\theta \\ &+ \sum\_{m=-\infty}^{\infty} \sum\_{n=0}^{\infty} \cos p\_{n} (h-d\_{1}) \int\_{0}^{2\pi} \int\_{0}^{R\_{1}} A\_{mn}^{(3,p)} \mathcal{I}\_{mn}(p\_{n}r) \mathrm{e}^{im\theta} r \mathrm{d}r \mathrm{d}\theta \end{split} \tag{72}$$

*f* (1,*p*) <sup>55</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>1</sup> *z* − *zG*<sup>1</sup> *φ*(5,*p*) *R*2 ( ( ( *r*=*R*<sup>1</sup> **<sup>n</sup>***x*·(−**ns**<sup>1</sup> )*R*1d*<sup>z</sup>* <sup>−</sup> \* *<sup>R</sup>*<sup>1</sup> <sup>0</sup> *<sup>φ</sup>*(5,*p*) *R*1 ( ( ( *z*=−*d*<sup>1</sup> (*r* cos *θ*)**nz**·(**n***z*)*r*d*r* d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 *<sup>A</sup>*(5,*p*) *mn* <sup>I</sup>*mn*(*pnR*1)e*imθR*1**n***x*·(−**ns**<sup>1</sup> ) d*θ* \* 0 −*d*<sup>1</sup> *z* − *zG*<sup>1</sup> *Zn*(*z*) *Zn*(0)d*z* − *δ*1*<sup>p</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>1</sup> 0 −4(*h*−*d*1) <sup>2</sup>+*r*<sup>2</sup> <sup>8</sup>(*h*−*d*1) *<sup>r</sup>* cos *<sup>θ</sup>r*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* <sup>+</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 cos *pn*(*h* − *d*1) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>1</sup> <sup>0</sup> *<sup>A</sup>*(5,*p*) *mn* <sup>I</sup>*mn*(*pnr*)e*im<sup>θ</sup> <sup>r</sup>*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* % (73) *f* (1,*p*) <sup>15</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 <sup>−</sup>*d*<sup>1</sup> *<sup>φ</sup>*(5,*p*) *R*2 ( ( ( *r*=*R*<sup>1</sup> **n***x*·(−**ns**<sup>1</sup> )*R*1d*z*d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 *<sup>B</sup>*(5,*p*) *mn* <sup>I</sup>*mn*(*knR*1) <sup>+</sup> *<sup>C</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*knR*1) *<sup>R</sup>*1**n***x*·(−**ns**<sup>1</sup> )e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>1</sup> *Zn*(*z*) *<sup>Z</sup>*0(*z*) <sup>d</sup>*<sup>z</sup>* (74) *f* (2,*p*) <sup>11</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>2</sup> *φ*(1,*p*) *R*2 ( ( ( *r*=*R*<sup>2</sup> **<sup>n</sup>***x*·(**ns**<sup>2</sup> )*R*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*(1,*p*) *R*4 ( ( ( *r*=*R*<sup>3</sup> **n***x*·(−**ns**<sup>3</sup> )*R*<sup>3</sup> % d*z*d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 &*B*(1,*p*) *mn* <sup>I</sup>*mn*(*knR*2) <sup>+</sup> *<sup>C</sup>*(1,*p*) *mn* <sup>K</sup>*mn*(*knR*2) *<sup>R</sup>*2**n***x*·(**ns**<sup>2</sup> ) <sup>+</sup> *<sup>F</sup>*(1,*p*) *mn* <sup>K</sup>*mn*(*R*3)*R*3**n***<sup>x</sup>* ·(−**ns**<sup>3</sup> )]e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> *Zn*(*z*) *Zn*(0)d*z* (75) *f* (2,*p*) <sup>51</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* <sup>0</sup> −*d*<sup>2</sup> *z* − *zG*<sup>2</sup> *φ*(1,*p*) *R*2 ( ( ( *r*=*R*<sup>2</sup> **<sup>n</sup>***x*·(**ns**<sup>2</sup> )*R*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*(1,*p*) *R*4 ( ( ( *r*=*R*<sup>3</sup> **n***x*·(−**ns**<sup>3</sup> )*R*<sup>3</sup> % d*z* <sup>−</sup> \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> *<sup>φ</sup>*(1,*p*) *R*3 ( ( ( *z*=−*d*<sup>2</sup> (*r* cos *θ*)**nz**·(**n***z*)*r*d*r* % d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 *B*(1,*p*) *mn* <sup>I</sup>*mn*(*knR*2) <sup>+</sup> *<sup>C</sup>*(1,*p*) *mn* <sup>K</sup>*mn*(*knR*2) *<sup>R</sup>*2**n***x*·(**ns**<sup>2</sup> ) <sup>+</sup> *<sup>F</sup>*(1,*p*) *mn* <sup>K</sup>*mn*(*knR*3)*R*3**n***<sup>x</sup>* ·(−**ns**<sup>3</sup> )}e*imθ*d*<sup>θ</sup>* \* 0 −*d z* − *zG*<sup>2</sup> *Zn*(*z*) *Zn*(0)d*z* <sup>−</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 cos *pn*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 *<sup>D</sup>*(1,*p*) *mn* <sup>I</sup>*mn*(*qnr*) <sup>+</sup> *<sup>E</sup>*(1,*p*) *mn* <sup>K</sup>*mn*(*qnr*) e*im<sup>θ</sup> r*<sup>2</sup> cos *θ*d*r*d*θ* (76) *f* (2,*p*) <sup>33</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> *<sup>φ</sup>*(3,*p*) *R*3 ( ( ( *z*=−*d*<sup>2</sup> **nz**·(**n***z*)*r*d*r*d*θ* = *δ*2*<sup>p</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 (*h*−*d*2) <sup>2</sup> <sup>−</sup> *<sup>r</sup>*<sup>2</sup> 4(*h*−*d*2) *r*d*r*d*θ* <sup>+</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 cos *pn*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 *<sup>D</sup>*(3,*p*) *mn* <sup>I</sup>*mn*(*qnr*) <sup>+</sup> *<sup>E</sup>*(3,*p*) *mn* <sup>K</sup>*mn*(*qnr*) e*im<sup>θ</sup> r*d*r*d*θ* (77) *f* (2,*p*) <sup>55</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>2</sup> *z* − *zG*<sup>2</sup> *φ*(5,*p*) *R*2 ( ( ( *r*=*R*<sup>2</sup> **<sup>n</sup>***x*·(**ns**<sup>2</sup> )*R*<sup>2</sup> <sup>+</sup> *<sup>φ</sup>*(5,*p*) *R*4 ( ( ( *r*=*R*<sup>3</sup> **n***x*·(−**ns**<sup>3</sup> )*R*<sup>3</sup> % <sup>d</sup>*<sup>z</sup>* <sup>−</sup> \* *<sup>R</sup>*<sup>3</sup> *<sup>R</sup>*<sup>2</sup> *<sup>φ</sup>*(5,*p*) *R*3 ( ( ( *z*=−*d*<sup>2</sup> (*r* cos *θ*)**nz**·(**n***z*)*r*d*r* d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 &*B*(5,*p*) *mn* <sup>I</sup>*mn*(*knR*2) <sup>+</sup> *<sup>C</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*knR*2) *<sup>R</sup>*2**n***x*·(**ns**<sup>2</sup> ) <sup>+</sup> *<sup>F</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*knR*3)*R*3**n***<sup>x</sup>* ·(−**ns**<sup>3</sup> )]e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> *z* − *zG*<sup>2</sup> *Zn*(*z*) *Zn*(0)d*z* − *δ*2*<sup>p</sup>* \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 −4(*h*−*d*2) <sup>2</sup>+*r*<sup>2</sup> <sup>8</sup>(*h*−*d*2) *<sup>r</sup>* cos *<sup>θ</sup>r*<sup>2</sup> cos *<sup>θ</sup>*d*r*d*<sup>θ</sup>* <sup>+</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 cos *pn*(*h* − *d*2) \* <sup>2</sup>*<sup>π</sup>* 0 \* *<sup>R</sup>*<sup>3</sup> *R*2 *<sup>D</sup>*(5,*p*) *mn* <sup>I</sup>*mn*(*qnr*) <sup>+</sup> *<sup>E</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*qnr*) e*im<sup>θ</sup> r*<sup>2</sup> cos *θ*d*r*d*θ* % (78) *f* (2,*p*) <sup>15</sup> <sup>=</sup> \* <sup>2</sup>*<sup>π</sup>* 0 \* 0 −*d*<sup>2</sup> *φ*(5,*p*) *R*2 ( ( ( *r*=*R*<sup>2</sup> **<sup>n</sup>***x*·(**ns**<sup>2</sup> )*R*<sup>1</sup> <sup>+</sup> *<sup>φ</sup>*(5,*p*) *R*4 ( ( ( *r*=*R*<sup>3</sup> **n***x*·(−**ns**<sup>3</sup> )*R*<sup>3</sup> % d*z*d*θ* <sup>=</sup> <sup>∞</sup> <sup>∑</sup>*m*=−<sup>∞</sup> ∞ ∑ *n*=0 \* <sup>2</sup>*<sup>π</sup>* 0 *B*(5,*p*) *mn* <sup>I</sup>*mn*(*knR*2) <sup>+</sup> *<sup>C</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*knR*2) *<sup>R</sup>*2**n***x*·(**ns**<sup>2</sup> ) <sup>+</sup> *<sup>F</sup>*(5,*p*) *mn* <sup>K</sup>*mn*(*knR*3)*R*3**n***<sup>x</sup>* ·(−**ns**<sup>3</sup> )}e*imθ*d*<sup>θ</sup>* \* 0 −*d*<sup>2</sup> *Zn*(*z*) *<sup>Z</sup>*0(*z*) <sup>d</sup>*<sup>z</sup>* (79)

Note that *f* (*q*,*p*) <sup>15</sup> and *f* (*p*,*q*) <sup>51</sup> are the same when the geometry of the floating body is symmetrical about the *x*-axis.

#### **7. Motion Responses of Floating Ring Structure**

Consider the horizontal coordinates of the centre of gravity of the floating regular polygonal ring structure coincide with the origin, i.e., *xG*<sup>1</sup> , *yG*<sup>1</sup> = *xG*<sup>2</sup> , *yG*<sup>2</sup> = (0, 0) but *zG*<sup>1</sup> = 0 and *zG*<sup>2</sup> = 0. As regular polygons are geometrically symmetrical about certain axes, the products of inertia of the floating regular polygonal platform/structure will become zero regardless of their orientation. If the motion of the floating body is relatively small, the following equations of motion in 3 DOFs (i.e., surge, heave and pitch) are satisfied.

$$
\begin{split}
\left[-\omega^{2}\left(m^{(1)} + \mu\_{11}^{(1,1)}\right) - i\omega\lambda\_{11}^{(1,1)}\right] \left\{\xi\_{1}^{(1)}\right\} + \left[-\omega^{2}\mu\_{15}^{(1,1)} - i\omega\lambda\_{15}^{(1,1)}\right] \left\{\xi\_{5}^{(1)}\right\} + \left[-\omega^{2}\mu\_{11}^{(1,2)} - i\omega\lambda\_{11}^{(1,2)}\right] \left\{\xi\_{1}^{(2)}\right\} \\ + \left[-\omega^{2}\mu\_{15}^{(1,2)} - i\omega\lambda\_{15}^{(1,2)}\right] \left\{\xi\_{5}^{(2)}\right\} - \left\{F\_{1}^{(1)}\right\}
\end{split}
\tag{80}
$$

$$
\omega \left[ -\omega^2 \left( m^{(1)} + \mu\_{33}^{(1,1)} \right) - i\omega\lambda\_{33}^{(1,1)} + \rho\chi A\_w^{(1)} \right] \left\{ \mathfrak{z}\_3^{(1)} \right\} + \left[ -\omega^2 \mu\_{33}^{(1,2)} - i\omega\lambda\_{33}^{(1,2)} \right] \left\{ \mathfrak{z}\_3^{(2)} \right\} = \left\{ F\_3^{(1)} \right\} \tag{81}
$$

$$
\begin{split}
\left[-\omega^{2}\mu\_{51}^{(1,1)} - i\omega\lambda\_{51}^{(1,1)}\right] \left\{\xi\_{1}^{(1)}\right\} + \left[-\omega^{2}\left(l\_{22}^{(1)} + \mu\_{56}^{(1,1)}\right) - i\omega\lambda\_{55}^{(1,1)} + m^{(1)}g\overline{\mathcal{L}\mathcal{M}}\_{\ast}^{(1)}\right] \left\{\xi\_{5}^{(1)}\right\} + \left[-\omega^{2}\mu\_{54}^{(1,2)} - i\omega\lambda\_{51}^{(1,2)}\right] \left\{\xi\_{1}^{(2)}\right\} \\ + \left[-\omega^{2}\mu\_{55}^{(1,2)} - i\omega\lambda\_{55}^{(1,2)}\right] \left\{\xi\_{5}^{(2)}\right\} - \left\{F\_{5}^{(1)}\right\}
\end{split} \tag{82}
$$

&

$$\begin{split} -\omega^{2} \left( m^{(2)} + \mu\_{11}^{(2,2)} \right) - i\omega \lambda\_{11}^{(2,2)} \left[ \left\{ \xi\_{1}^{(2)} \right\} + \left[ -\omega^{2} \mu\_{13}^{(2,2)} - i\omega \lambda\_{13}^{(2,2)} \right] \left\{ \xi\_{3}^{(2)} \right\} + \left[ -\omega^{2} \mu\_{11}^{(2,1)} - i\omega \lambda\_{11}^{(2,1)} \right] \left\{ \xi\_{1}^{(1)} \right\} \\ + \left[ -\omega^{2} \mu\_{13}^{(2,1)} - i\omega \lambda\_{13}^{(2,1)} \right] \left\{ \xi\_{3}^{(1)} \right\} - \left\{ F\_{1}^{(2)} \right\} \end{split} \tag{83}$$

$$
\begin{bmatrix}
\end{bmatrix}
\begin{Bmatrix}
\xi\_3^{(2)}
\end{Bmatrix} + \begin{bmatrix}
\end{Bmatrix}
\begin{Bmatrix}
\xi\_3^{(1)}
\end{Bmatrix} = \begin{Bmatrix}
F\_3^{(2)}
\end{Bmatrix}
\tag{84}
$$

$$
\begin{split}
\left[-\omega^{2}\mu\_{51}^{(2,2)} - i\omega\lambda\_{51}^{(2,1)}\right] \left\{\xi\_{1}^{(2)}\right\} &+ \left[-\omega^{2}\left(l\_{22}^{(2)} + \mu\_{55}^{(2,2)}\right) - i\omega\lambda\_{55}^{(2,2)} + m^{(2)}\rho\overline{G\mathcal{M}}\_{L}^{(2)}\right] \left\{\xi\_{5}^{(2)}\right\} + \left[-\omega^{2}\mu\_{54}^{(2,1)} - i\omega\lambda\_{51}^{(2,1)}\right] \left\{\xi\_{1}^{(1)}\right\} \\ &+ \left[-\omega^{2}\mu\_{55}^{(2,1)} - i\omega\lambda\_{55}^{(2,1)}\right] \left\{\xi\_{5}^{(1)}\right\} = \left\{F\_{5}^{(2)}\right\}
\end{split}
\tag{85}
$$

where *m*(*p*) is the mass of the oscillating body *p*, *ξ* (*p*) *<sup>i</sup>* the unknown displacement of the oscillating body *<sup>p</sup>* at *<sup>i</sup>*-th radiation mode, *<sup>F</sup>*(*q*) *<sup>i</sup>* the wave exciting force acting on the floating body *q*, *I* (*p*) *ii* (*i* = 1, 2, 3) the mass moment of inertia of the oscillating body *p* about *i*-axis, *<sup>A</sup>*(*p*) *<sup>w</sup>* the waterplane area of the oscillating body *<sup>p</sup>*, *<sup>V</sup>*(*p*) *<sup>w</sup>* the wetted volume of the oscillating body *<sup>p</sup>*, *GM*(*p*) *T* <sup>=</sup> *<sup>A</sup>*(*p*) *<sup>w</sup>*<sup>11</sup> *<sup>V</sup>*(*p*) *<sup>w</sup>* + *z* (*p*) *<sup>B</sup>* − *z* (*p*) *G* the transverse metacentric height of the oscillating body *<sup>p</sup>*, *GM*(*p*) *L* <sup>=</sup> *<sup>A</sup>*(*p*) *<sup>w</sup>*<sup>22</sup> *<sup>V</sup>*(*p*) *<sup>w</sup>* + *z* (*p*) *<sup>B</sup>* − *z* (*p*) *G* the longitudinal metacentric height of the oscillating body *<sup>p</sup>* and *<sup>A</sup>*(*p*) *wii* (*<sup>i</sup>* <sup>=</sup> 1, 2) is the second moment over the waterplane area of the oscillating

body *p* about *i*-axis. (*p*)

By linear algebra, the motion responses *ξ <sup>i</sup>* (*i* = 1, 2, ··· , 6) are solved by using Equations (80)–(85). By substituting the solutions into Equation (4), one can obtain the radiated potential.

#### **8. Verification of Semi-Analytical Approach and Computer Code**

In order to verify the semi-analytical approach and the computer code, we compare the hydrodynamic results (i.e., added mass, radiation damping, wave exciting forces, RAO and wave field) with those obtained from the commercial software ANSYS AQWA based on the Boundary Element Method.

For the verification exercise, we consider a floating hexagonal platform that is placed within a floating hexagonal ring structure whose geometries are defined by the radius function as given in Equation (1) with *R*<sup>01</sup> = 50 m, *R*<sup>02</sup> = 90 m, *R*<sup>03</sup> = 100 m, *ε*1,2,3 = 0.03, *np*1,2,3 = 6 and *<sup>θ</sup>*01,2,3 = *<sup>π</sup>* <sup>6</sup> . The drafts *d*<sup>1</sup> for the platform and *d*<sup>2</sup> for the ring structure are equally 10 m. Figure 3 shows the structural shape. It is assumed that the centre of gravity coincides with the origin and the water depth is 50 m. The incident wave is along the *x*-axis (i.e., *β* = 0◦).

**Figure 3.** Hexagonal ring structure for verification study: (**a**) plan view; (**b**) 3D mesh model (AQWA).

For the 3D model in AQWA, the maximum mesh size is 2.8 m which leads to a total of 11,459 diffracting elements to be used. It should be noted that a fine mesh model is required to obtain a reasonably converged inner free water elevation. The regular wave angular frequency domain for AQWA is divided into 20 frequencies for the interval [0.3866 rad/s, 1.2528 rad/s], which is equivalent to *kh* = [1, 8].

Figure 4 presents the added mass and radiation damping obtained from the present semi-analytical method and AQWA. The numbers of truncated terms for the present semianalytical method were taken as *M* = 30, *N* = 5, *Nr* = 12 and *Nq* = 24 for parametric studies. It is unnecessary to perform the integration for all the *m*-series terms, but series index *m* can be restricted to

$$\begin{array}{c} m = \pm 1, \ \pm (n\_p \pm 1) \quad \text{for surely or sway} \\ m = 0, \ \pm n\_p \quad \text{for have} \\ m = \pm 1, \ \pm (n\_p \pm 1), \ \pm (2n\_p \pm 1) \quad \text{for roll or pitch} \\ m = 0, \ \pm 2, \ \pm n\_p, \ \pm (n\_p \pm 2), \ \pm (2n\_p \pm 1), \ \pm 2n\_p \quad \text{for yaw} \end{array} \tag{86}$$

where *np* is the parameter of the radius function as given in Equation (1). Also, it should be noted that *Nr* and *Nq* can be taken as multiple numbers of *np*. For instance, *Nr* = 12 and *Nq* = 24 are respectively 2 and 4 times *np* = 6 for a hexagonal shape. This rule can be equally applied for calculating the wave exciting forces. Hence, such a selective calculation can speed up the hydrodynamic analysis in composing the global system matrix as well as obtaining the wave exciting forces and radiation forces.

**Figure 4.** *Cont*.

**Figure 4.** Comparison of normalised added mass and radiation damping obtained from AQWA and present approach: (**a**) added mass for surge; (**b**) added mass for heave; (**c**) added mass for pitch; (**d**) added mass for surge-pitch; (**e**) radiation damping for surge; (**f**) radiation damping for heave; (**g**) radiation damping for pitch; (**h**) radiation damping for surge-pitch.

The added mass was normalised by

$$\overline{\mu}\_{11}^{(q,p)} = \frac{\mu\_{11}^{(q,p)}}{\rho \mathbb{S}\_{0q} d\_q}; \overline{\mu}\_{33}^{(q,p)} = \frac{\mu\_{33}^{(q,p)}}{\rho \mathbb{S}\_{0q} d\_q}; \overline{\mu}\_{55}^{(q,p)} = \frac{\mu\_{55}^{(q,p)}}{\rho \mathbb{S}\_{0q}^2 d\_q}; \overline{\mu}\_{15}^{(q,p)} = \frac{\mu\_{15}^{(q,p)}}{\rho \mathbb{S}\_{0q}^{1.5} d\_q} \tag{87}$$

while the radiation damping was normalised by

$$\overline{\lambda}\_{11}^{(q,p)} = \frac{\lambda\_{11}^{(q,p)}}{\rho \omega \mathbb{S}\_{0\_q} d\_q}; \overline{\lambda}\_{33}^{(q,p)} = \frac{\lambda\_{33}^{(q,p)}}{\rho \omega \mathbb{S}\_{0\_q} d\_q}; \overline{\lambda}\_{55}^{(q,p)} = \frac{\lambda\_{55}^{(q,p)}}{\rho \omega \mathbb{S}\_{0\_q}^2 d\_q}; \overline{\lambda}\_{15}^{(q,p)} = \frac{\lambda\_{15}^{(q,p)}}{\rho \omega \mathbb{S}\_{0\_q}^{1.5} d\_q} \tag{88}$$

where *S*0*<sup>q</sup>* is the cross-sectional area of the polygonal platform for *q* = 1 and that of the polygonal ring structure for *q* = 2. The present semi-analytical results and AQWA results are in excellent agreement; thereby confirming the validity, convergence and accuracy of the semi-analytical approach. It can be seen that the magnitudes of crests near resonance frequencies are sensitively varying with minor discrepancies between AQWA and present results (see Figure 4b,d).

The wave exciting forces and RAOs were calculated for *β* = 0◦. The horizontal wave force, vertical wave force and rotational moment were normalised as follows:

$$\overline{F\_x}^{(q)} = \frac{F\_x^{(q)}}{\rho g A S\_{0\_q}};\\ \overline{F\_z}^{(q)} = \frac{F\_z^{(q)}}{\rho g A S\_{0\_q}};\\ \overline{M}\_y^{(q)} = \frac{M\_y^{(q)}}{\rho g A S\_{0\_q} d\_q}$$

The RAO (Response Amplitude Operator) is defined as the motion response normalised by the incident wave amplitude *A*. Figure 5 compares the wave exciting forces and the RAOs obtained from the present semi-analytical method and those computed from ANSYS AQWA. It can be seen that the results are in close agreement; thereby verifying the present semi-analytical formulation and method of solution.

**Figure 5.** Comparison of wave exciting forces and RAOs obtained from AQWA and present approach: (**a**) surge force; (**b**) heave force; (**c**) pitch moment; (**d**) RAO for surge; (**e**) RAO for heave; (**f**) RAO for pitch.

The wave field and profiles for the inner water basin of the floating hexagonal platform and ring structure were calculated as shown in Figure 6 (for *T* = 10 s). The floating bodies are oscillating together or individually. Wave profiles are drawn along the *x*-axis at *y* =0m and *y* = 50 m, which are normalised by the incident wave amplitude for the wave period 10 s. The wave profiles obtained from the present approach and AQWA are well matched. Note that the wave profiles belonging to AQWA were extracted from AQWA FLOW by writing a computer code to process the raw data at selected points.

**Figure 6.** Comparison of normalised wave elevations and profiles for together or individually oscillating floating hexagonal platform and ring structure obtained from AQWA and present approach (for *T* = 10 s): (**a**) wave field for monolithic motion; (**b**) wave field for individual motion; (**c**) wave profile for monolithic motion; (**d**) wave profile for individual motion.
