Effect of articulations on the hydroelastic analysis of VLFS integrated with porous floating breakwater

Abstract

The present study examines the hydroelastic interactions of a multi-module very large floating structure interconnected by articulated hinges and integrated with porous floating breakwaters. Unlike monolithic VLFS designs, the articulated VLFS introduce unique dynamic challenges due to connector forces and inter-module wave interference, which have not been thoroughly explored when integrated with porous wave attenuation mechanisms. The study develops a coupled numerical framework that combines a multi-domain boundary element method for wave interaction with porous structures and a finite difference method for the structural response of hinged modules. This model explicitly resolves interactions between waves, structures, and connectivity, including the effects of breakwater porosity and hinge constraint dynamics. Critical parameters such as hinge stiffness, module spacing, breakwater placement, and porosity are systematically varied to quantify their influence on inter-modular bending moments, shear forces, and wave transmission. The investigation demonstrate that articulation redistributes peak stresses, reducing maximum bending moments in individual modules by 15%–25% compared to rigid configurations, although it can amplify torsion at hinge points under oblique waves. Moreover, porous breakwaters that are strategically positioned between modules suppress wave resonance in the gaps, reducing connector forces by 30%–40% and attenuates transmitted energy by over 45%. However, to avoid detrimental wave trapping between modules, the porosity of the breakwaters must exceed 25%. The study establishes that optimal hinge design, combined with breakwaters tailored for porosity, mitigates hydroelastic penalties in multi-module systems which enables scalable VLFS deployments in exposed seas and advanced design methodologies for adaptive, large-scale floating infrastructure.

Introduction

The design of very large floating structures (VLFS) for offshore urbanization and renewable energy hubs poses a significant engineering challenge due to mitigating their vulnerability to wave-induced loads while ensuring structural integrity across vast, flexible areas. Early monolithic designs, which prioritized continuous buoyancy, often failed when subjected to wave action. The introduction of hydroelasticity theory showed that structural flexibility could help manage hydrodynamic forces, but it also highlighted the need for more advanced solutions. To enhance resilience, a key engineering strategy is the incorporation of articulated hinges that segment VLFS into modular units. However, this approach introduces a fundamental design trade-off: while these discontinuities reduce global bending moments, they can unintentionally increase localized stress concentrations and trigger complex resonance modes during extreme sea conditions. Additionally, although porous floating breakwaters are effective for wave attenuation, their interaction with the dynamic responses created by articulations remains largely unexplored. As climate change intensifies wave conditions, optimizing the relationship between articulation configuration, breakwater efficiency, and hydroelastic response becomes an essential engineering priority. The implications of the study are significant for addressing critical challenges in polar coastal protection and offshore engineering. Noteworthy contributions in this domain include studies by Hosking et al., ¹ Meylan and Squire, ² Meylan, ³ Porter and Porter, ⁴ Meylan et al., ⁵ and Ren et al. ⁶

Recent research on VLFS has focused on understanding their hydrodynamic and hydroelastic properties across various models. A key area of investigation is the influence the structural response of the VLFS due to the articulations and engineered hinge points. The evaluations emphasize the importance of flexible connections, mooring configurations, and complex marine forces. The findings highlight that the behavior of the structures in water, combined with the design of the articulations, fundamentally determines the dynamics and operational stability of VLFS. When deployed in deep water, VLFS depend heavily on functional hinge systems. These hinge points are particularly vulnerable to severe structural damage, leading to high repair and maintenance costs. Ohkusu and Namba ⁷ presented an analytical method to investigate the hydroelastic response of very large floating structures in shallow water depth. The findings demonstrated that the proposed method accurately predicted wave–structure interaction and elastic deformation when compared with more computationally intensive numerical models. Hong and Hong ⁸ analyzed the hydroelastic responses and time-mean horizontal drift forces of a VLFS when connected to a floating OWC breakwater system via a pin connection. The proposed analytical method and numerical results provide valuable insights for designing pin-connected breakwater systems to mitigate the hydroelastic responses of VLFS. Porter and Evans ⁹ examined the diffraction of flexural waves in a thin elastic sheet floating on water, focusing specifically on the effect of finite straight crack. The study provided important insights into wave–crack interactions, offering a theoretical foundation for assessing crack propagation and failure in floating elastic structures such as sea ice sheets or pontoon-type VLFS. Wu et al. ¹⁰ conducted numerical simulations for five cases with varying underwater leg lengths using three-dimensional potential theory. The findings indicate that the presence and length of the underwater legs significantly affect the vessel’s hydrodynamic characteristics and responses, with the exception of heave motion. Cheng et al. ¹¹ employed a modal expansion approach to analyze fluid–structure interaction in the frequency domain. The study suggests that the VLFS deflection is reduced by enhancing wave energy dissipation and adding damping. Singla et al. ¹² investigated the effectiveness of a floating porous plate in reducing the wave-induced structural response of VLFS using the eigenfunction expansion method. The study concludes that strategically placed floating porous plates can effectively decrease the hydroelastic response of VLFS by dissipating wave energy and influencing wave reflection and transmission patterns. Zheng et al. ¹³ demonstrated that hydrodynamic interactions can enhance wave power dissipation, highlighting the potential advantages of porous elastic plates for wave power extraction. The findings from these studies are applicable to various floating structures, particularly in modeling energy loss in flexible ice floes and for flexible plate wave energy converters.

The numerical simulation of fluid–structure interaction (FSI) phenomena in VLFS necessitates advanced computational strategies to concurrently address fluid dynamics and structural mechanics. The inviscid fluid model, based on potential flow theory, provides enhanced computational efficiency and is well-suited for applications where viscous dissipation is secondary. To model the resulting hydroelastic responses, integrated boundary element method (BEM) and finite element method (FEM) approaches have gained prominence as effective solutions. Zheng et al. ¹⁴ presented an effective BEM for analyzing the interaction between oblique waves and long prismatic structures in bodies of water with finite depth. The efficiency and accuracy of this method have been validated through comparisons of numerical results related to hydrodynamic coefficients and wave forces with other analytical and numerical techniques. Belibassakis ¹⁵ investigated a hybrid technique for conducting hydrodynamic analyses of floating bodies in areas with variable bathymetry. On comparing results with benchmark solutions, the study assessed the convergence and accuracy of the method in three dimensions. Riyansyah et al. ¹⁶ focused on the design of connections for a two-floating-beam system to minimize hydroelastic response. The findings revealed that the compliance of the two-floating beam system is influenced by the location and rotational stiffness of the connections. Koley ¹⁷ examined the scattering of obliquely incident gravity waves by a horizontal, floating, flexible porous plate in finite-depth water with a variable bottom profile. The study concluded that a protrusion-type bed profile is most effective for creating a tranquillity zone in the confined space between the plate and a rigid wall. The proposed solution methodology is noted to be computationally efficient and applicable to more complex issues encountered in ocean engineering. Mohapatra and Soares ¹⁸ developed a hydroelastic model for the problem of linear wave interaction with a submerged horizontal flexible porous structure in three-dimensions. The accurateness of the three-dimensional solution is inspected and the hydroelastic behavior of the submerged porous structure is analyzed for oscillation modes, mooring stiffness, and porous-effect parameters on numerical assessment of reflection, dissipation coefficients, and plate displacements in different cases. Thereafter, Mohapatra and Soares ¹⁹ developed a 3D hydroelastic model subjected to linear wave interaction with horizontal flexible floating and submerged porous structures and observed that the analysis could be useful for a greater insight to design a multi-use flexible floating system. Recently, Amouzadrad et al. ²⁰ established a novel analytical model to investigate wave-current interaction with a moored, interconnected, finite floating structure and found that the connectors’ mechanics and wave-current conditions affect the hydroelastic response of the floating interconnected platform.

Investigations into the hydroelastic behavior of VLFS and neighboring floating structures under the influence of articulated VLFS have been conducted in recent works (Jiang et al., ²¹ Cheng et al., ²² Wei et al., ²³ Fang et al., ²⁴ Wang et al., ²⁵ and Mohapatra et al. ²⁶ ). Sahoo et al. ²⁷ proposed a model consisting of either a bottom-standing porous structure or a surface-piercing porous structure, combined with an elastic plate placed apart from one another to protect a seawall. Wei et al. ²⁸ conducted an experimental investigation focusing on the hydrodynamic characteristics of a proposed modular floating structure system integrated with WEC-type floating artificial reefs. The findings of the work provide valuable references for engineering applications involving modular floating structures integrated with WECs. Zhu et al. ²⁹ examined the hydrodynamic responses of marine trains composed of multiple carriages under different connection types, offering valuable insights for designing this joint system to enhance its engineering applications. The numerical results indicated that the motion response amplitude operators (RAOs) of the vessels significantly increased with higher advancing speeds, remaining relatively unaffected by changes in joint position and the gap between ships. Wang et al. ³⁰ utilized a mooring system to secure the position and heading of a very large floating structure (VLFS) against the forces of waves, wind, and currents. The results showed that the general connector model is reasonable and applicable to various types of connectors, thus facilitating an analysis of the mechanical properties of the connectors influence the motion response of the VLFS.

The research highlights the urgent need to improve hydroelastic models for VLFS, especially in areas that require large-scale modular deployment. This involves thoroughly considering complex articulation mechanics. Engineered hinge points and flexible joints play a crucial role in the interaction of waves with the structure, functioning as essential mechanisms for redistributing loads and dissipating energy (Jiang et al. ²¹ ). Neglecting articulation dynamics during the design phase can lead to operational challenges, as it increases the risk of localized joint failures and overall structural issues. On the other hand, including hinge performance in hydroelastic frameworks significantly decreases long-term structural risks. When relative motions at articulations are not managed, it can lead to hinge fatigue, buckling, or rupture failures. These failures require complex repairs, recertification, and even module replacements, which disrupt core platform functions and lead to extensive operational downtime. Therefore, prioritizing the detailed analysis and optimization of articulation systems ensures both the structural integrity and continuous functionality of multi-module VLFS. This approach aligns operational reliability with hydroelastic performance across diverse marine environments. The study aims to bridge that gap through a systematic investigation of VLFS configured with one, two, and three articulations, integrated with porous breakwaters. The findings establish new design paradigms that transform articulations from potential liabilities into assets for resilience, thereby extending the operability of VLFS in turbulent ocean conditions.

The present study introduces the hydroelastic analysis of articulated multi-module VLFS integrated with porous floating breakwaters, an essential configuration for scalable offshore expansion. Unlike previous research focused on rigidly connected or single-module VLFS, the study uniquely explores the tripartite coupling between wave connectivity dynamics, inter-modular fluid resonance, and wave attenuation mediated by breakwaters. The advanced computational tool is developed that extends the MDBEM for porous wave interactions (Patil and Karmakar ³¹ ) and integrates it with the FDM. This framework allows to simultaneously resolve the kinematics of articulated modules, connector forces, and fluid-porosity-structure interactions. The MDBEM approach explicitly models hinge constraints, gap wave trapping, and porosity-dependent energy dissipation between modules, thereby overcoming the limitations of conventional monolithic VLFS models. Key parameters influencing system performance include hinge stiffness, module spacing, breakwater porosity, and the spatial arrangement concerning wave incidence. The findings quantify the redistribution of hydroelastic loads due to articulation, analyze the position of breakwaters in suppressing resonant responses in inter-modular gaps, and reveal optimal strategies to minimize connector fatigue while maximizing wave energy dissipation. The outcomes of this study establish the design protocols for resilient, reconfigurable floating infrastructure in exposed marine environments. The study directly supports next-generation applications, such as modular floating cities, adaptive renewable energy platforms, and dynamic coastal defenses, where the interactions between waves, structures, and connectivity determine structural viability.

Theoretical formulation

The hydrodynamic interaction of ocean waves with an articulated VLFS incorporating a porous floating box is analyzed using small-amplitude wave theory. The configuration, depicted in Figure 1, is represented mathematically as a boundary value problem (BVP) within a two-dimensional coordinate system. A monochromatic incident wave propagates in the positive x - direction. The fluid domain is of infinite horizontal extent $-\infty <x<\infty$ over a constant water depth $0<y<h$ . The VLFS is modeled as a two-dimensional structure oriented along the x - axis, occupying the spatial interval $0<x<L$ , where $\prime L\prime$ is the length of the VLFS, the y - axis is directed vertically downward. The porous box is rigidly attached to the VLFS. Assuming the fluid flow is incompressible, inviscid, irrotational, and simple harmonic in time with angular frequency ω, the velocity potential follows the relation $\varphi (x,y,z,t)$  =  $\mathrm{Re}\{\varphi (x,y)e^{i(k_z-\omega t)}\}$ . Here, $k_z=k\sin \theta$ represents the spatial complex potential, and ‘Re’ denotes the real part. This potential satisfies the governing Helmholtz equation across the entire fluid domain, denoted by:

$$({\nabla }^2-{k_z}^2)\varphi (x,y)=0\mathrm{in}\mathrm{the}\mathrm{fluid}\mathrm{domain}\Omega .$$

(1)

The free surface boundary ${\mathrm{\Gamma }}_{\mathrm{FS}}$ is presented by:

$$\frac{\partial \varphi }{\partial y}-\frac{{\omega }^2}{g}\varphi =0\mathrm{on}{\mathrm{\Gamma }}_{\mathrm{FS}}\mathrm{at}\mathrm{y}=0.$$

(2)

The seabed boundary ${\mathrm{\Gamma }}_{\mathrm{SB}}$ is assumed to be impervious, so the seabed boundary condition is denoted by:

$$\frac{\partial \varphi }{\partial y}=0,\mathrm{on}{\mathrm{\Gamma }}_{\mathrm{SB}}\mathrm{at}y=-h.$$

(3)

The Sommerfeld radiation condition specifies the input and output boundaries, defined by ${\mathrm{\Gamma }}_{\mathrm{IR}}$ and ${\mathrm{\Gamma }}_O$ , respectively. Here, ‘IR’ represents the incident and reflected wave components, while ‘O’ denotes the transmitted waves. The corresponding boundary conditions are indicated by:

$$\underset{r\to \infty }{\lim }\sqrt{r}(\frac{\partial \varphi }{\partial r}-\mathrm{ik}\varphi )=0,\mathrm{on}{\mathrm{\Gamma }}_{\mathrm{IR}}\mathrm{and}{\mathrm{\Gamma }}_O,$$

(4)

Here, $r=\sqrt{x^2+y^2}$ , using the idea of superposition of waves, the velocity potential at input and output boundary conditions can be written by:

$$\begin{array}{cc}\multicolumn{1}{c}{}& \mathrm{Input}\mathrm{region}({\mathrm{\Gamma }}_{\mathrm{IR}}):{\varphi }_{\mathrm{IR}}={\varphi }_I+{\varphi }_R,\hfill \\ \multicolumn{1}{c}{}& \mathrm{Output}\mathrm{region}({\mathrm{\Gamma }}_O):{\varphi }_O={\varphi }_T,\hfill \end{array}$$

(5)

Here, ϕ _I represents the incident wave potential, ϕ _R the reflected wave potential, and ϕ _T the transmitted wave potential. As derived from small-amplitude wave theory, the velocity potentials in the input and output regions are given by:

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\varphi }_I=(I_0{e}^{\mathrm{ik}\cos \theta }+R_0{e}^{-\mathrm{ik}\cos \theta })P_w,\hfill \\ \multicolumn{1}{c}{}& {\varphi }_O=T_0{e}^{\mathrm{ik}\cos \theta }{P}_w,\hfill \end{array}$$

(6)

where, the incident wave amplitude is denoted by I₀, the reflected wave amplitude by R₀, and the transmitted wave amplitude by R₀ at the input and output boundaries, respectively.

Figure 1.

Two-dimensional schematic representation of oblique wave interaction with a multi-module VLFS.

The wave pressure coefficient P _w , defined as a function of incident wave height H, acceleration due to gravity H, angular frequency ω, and pressure response factor P _p , is mathematically represented by the expression:

$$p_w=\frac{\mathrm{igH}}{2\omega }{p}_p.$$

(7)

The pressure response factor P _p is directly proportional to the water depth h and is denoted by $P_p=\frac{\cosh k(h+y)}{\cosh \mathrm{kh}}$ , where k is the wave number.

The model considers a wave train incident at an angle relative to the normal of the structure. The 2D representation in the x-z plane corresponds to a cross-section along the wave propagation direction, with the out-of-plane (y-direction) dependence analytically accounted for through the separation of variables in the velocity potential, as detailed in Section 2 –equations (20) and (21). This approach is standard for analyzing the hydroelastic response of long, prismatic structures to oblique waves.

The angular frequency ω satisfies the dispersion relation on the mean free surface is given by:

$${\omega }^2=\mathrm{gk}\tanh \mathrm{kh}.$$

(8)

The boundary conditions for a rigid structure are fundamentally determined by its permeability. In the case of a rigid, impermeable structure, the no-flow condition is mathematically expressed by:

$$\frac{\partial \varphi }{\partial n}=0,\mathrm{on}{\mathrm{\Gamma }}_{\mathrm{STR}}.$$

(9)

For a rigid permeable structure, the governing boundary condition incorporates the physical assumptions of continuous pressure and uninterrupted mass flux across the fluid–structure interface, mathematically expressed as:

$$\frac{\partial {\varphi }_{\mathrm{STR}}^f}{\partial n}=-\epsilon \frac{\partial {\varphi }_{\mathrm{STR}}^s}{\partial n}\mathrm{and}{\varphi }_{\mathrm{STR}}^f=(S+\mathrm{if}){\varphi }_{\mathrm{STR}}^s,$$

(10)

Here, the velocity potential in the water region is designated as ${\varphi }^f$ , and the velocity potential inside the porous structure region is denoted by ${\varphi }^f$ . The porosity parameter of the floating breakwater is represented by ε. Additionally, f signifies the linearized friction coefficient, and S corresponds to the inertia coefficient of the porous breakwater, as established in the foundational works of Solitt and Cross ³² and Dalrymple et al. ³³ These friction and inertia coefficients are determined through the mathematical relation expressed by:

$$f=\frac{1}{\omega }\frac{{\int }_V\mathrm{dV}{\int }_t^{t+T_w}{\epsilon }^2(\frac{\nu q^2}{K_p}+\frac{C_f\epsilon }{\sqrt{K_p}}{\left|\vartheta \right|}^3)\mathrm{dt}}{{\int }_V\mathrm{dV}{\int }_t^{t+T_w}\epsilon {\vartheta }^2\mathrm{dt}},$$

(11)

$$S=1+A_m\left(\frac{1-\epsilon }{\epsilon }\right),$$

(12)

where, virtual added mass on the porous structure resulting from wave impact is given as A _m , while ϑ represents the instantaneous Eulerian velocity vector at any spatial point. The parameter ν denotes the kinematic viscosity, and K _p signifies the intrinsic permeability. The volume is represented by V, with C _f being a dimensionless turbulent resistance coefficient and T _w indicating the wave period. In the current investigation, parameters S and f maintain identical definitions to those established in Lee. ³⁴ The boundary conditions for the flexible elastic VLFS, these are formulated in terms of vertical displacement and deflection amplitude. The vertical displacement of the floating elastic VLFS is mathematically shown as $\xi (x,t)=\mathrm{Re}\{\eta (x)e^{-i\omega t}\}$ , where $\eta (x)$ corresponds to the deflection amplitude. Further, the dynamic pressure P _d acting upon the floating elastic VLFS is given by:

$$\mathrm{EI}{(\frac{{\partial }^2}{\partial x^2}-k_z^2)}^2\eta -m_p{\omega }^2\eta =p_d\mathrm{at}0<x<L,y=0,$$

(13)

The flexural rigidity of the VLFS is denoted by EI, while m _p represents the mass per unit length of the floating VLFS. Additionally, the hydrodynamic pressure P _h acting on the wave-structure interface is defined by:

$$P_h=i\rho \omega \varphi -\rho g\eta \mathrm{on}y=0.$$

(14)

On assuming $P_d=P_h$ at y= 0, the thin plate equation shown in equation (13) will be of the form:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}{(\frac{{\partial }^2}{\partial x^2}-k_z^2)}^2\eta +(\rho g-m_p{\omega }^2)\eta =i\rho \omega \varphi }\\ \multicolumn{1}{c}{\mathrm{at}0<x<L,y=0.}\end{array}$$

(15)

The kinematic boundary condition on the floating structure is shown as:

$$\frac{\partial \varphi }{\partial y}=-i\omega \eta (x),0<x<L,y=0.$$

(16)

The mooring lines are connected to the elastic VLFS specifically at its free edges. Consequently, the bending moment and shear force boundary conditions arising from the mooring stiffness are mathematically represented by:

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}\frac{\partial }{\partial x}\{\frac{{\partial }^2}{\partial x^2}-(2-\nu )k_z^2\}\eta =Q_j\frac{\partial \eta }{\partial x}\mathrm{and}(\frac{{\partial }^2}{\partial x^2}-\nu k_z^2)}\\ \multicolumn{1}{c}{\eta =0\mathrm{at}x=0,L}\end{array}$$

(17)

where Q _j is the mooring stiffness.

The VLFS is considered to be articulated using the vertical linear springs of stiffness k₃₃ and flexural rotational springs of stiffness k₅₅ (Xia et al., ³⁵ and Karmakar et al.^36,37) is given by,

$$\begin{array}{c}\multicolumn{1}{c}{-\mathrm{EI}\frac{\partial }{\partial x}(\frac{{\partial }^2}{\partial x^2}-k_z{l}^2){\eta }_{+}=k_{33}({\eta }_{+}-{\eta }_{-}),}\\ \multicolumn{1}{c}{-\mathrm{EI}\frac{\partial }{\partial x}(\frac{{\partial }^2}{\partial x^2}-k_z{l}^2){\eta }_{-}=k_{33}({\eta }_{+}-{\eta }_{-})}\end{array}$$

(18)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{EI}(\frac{{\partial }^2}{\partial x^2}-k_z{l}^2){\eta }_{+}=k_{55}(\frac{\partial {\eta }_{+}}{\partial x}-\frac{\partial {\eta }_{-}}{\partial x}),}\\ \multicolumn{1}{c}{\mathrm{EI}(\frac{{\partial }^2}{\partial x^2}-k_z{l}^2){\eta }_{-}=k_{55}(\frac{\partial {\eta }_{+}}{\partial x}-\frac{\partial {\eta }_{-}}{\partial x})}\end{array}$$

(19)

where, ${\eta }_{+}$ denotes the joint of the first VLFS unit and ${\eta }_{-}$ denotes the joint of the second VLFS unit.

Method of solution

This section employs the Multi-Domain Boundary Element Method (MDBEM) to analyze the wave-structure interactions involving a VLFS integrated with multiple porous breakwaters. The MDBEM approach is built upon two core components: the fundamental solution and Green’s second identity. To solve the boundary value problem, the weighted residue technique and the Green-Gauss theorem are applied to the governing Helmholtz equation. The fundamental solution for this governing equation satisfies the mathematical relation expressed by:

$${\nabla }^{2}G-{k_z}^{2}G=\delta (\xi -x,\zeta -y),$$

(20)

where, the free surface Green’s function G is represented by:

$$G(x,y,\xi ,\zeta )=\frac{-K_0(k_z,r)}{2\pi }.$$

(21)

K₀ represents the modified zeroth-order Bessel function of the second kind, while $r=\sqrt{{(x-\xi )}^2+{(y-\eta )}^2}$ represents the distance between the field point and source point. The normal derivative of Green’s function is given by:

$$\frac{\partial G}{\partial n}=\frac{k_z}{2\pi }{K}_1\left(k_{z}r\right)\frac{\partial r}{\partial n},$$

(22)

where K₁ is a modified first-order Bessel function of the second kind, and in the case of singularity $r\to 0$ , the asymptotic behavior of K₀ is given by:

$$K_0\left(k_{z}r\right)=-\gamma -\ln \left(\frac{k_{z}r}{2}\right),$$

(23)

where, γ= 0.5772 is the Euler’s constant. Furthermore, for normal incidence, when θ= 0, the $K_0\left(k_{z}r\right)$ tends to approach $-\ln \left(r\right)$ . The computational domain is defined by the general field point $P(x,y)$ and source point $Q(\xi ,\zeta )$ . The free-surface Green’s function, serves as the fundamental solution for the Helmholtz equation, is numerically shown in equation (20). The generalized formulation of the boundary integral equation, derived on application of Green’s second identity to the free-space Green’s function, is given by:

$$c(P)\varphi \left(P\right)+{\int }_{\mathrm{\Gamma }}\varphi \left(Q\right)\frac{\partial G(P,Q)}{\partial n}d\mathrm{\Gamma }={\int }_{\mathrm{\Gamma }}G(P,Q)\frac{\partial \varphi \left(Q\right)}{\partial n}d\mathrm{\Gamma },$$

(24)

where, c(P) is given by:

$$c(P)=\{\begin{array}{c}1\mathrm{if}\mathrm{P}\in \Omega ,\hfill \\ \multicolumn{1}{c}{\frac{1}{2}\mathrm{if}\mathrm{P}\in \mathrm{\Gamma }\mathrm{is}\mathrm{smooth},}\\ \multicolumn{1}{c}{0\mathrm{if}\mathrm{P}\in \Omega ,\mathrm{\Gamma },}\end{array}$$

(25)

where, Γ denotes the considered boundary of the computational domain. Hence, the final boundary integral equation can be represented as:

$$\frac{\varphi \left(P\right)}{2}+{\int }_{\mathrm{\Gamma }}\frac{\partial G(P,Q)}{\partial n}\varphi \left(Q\right)d\mathrm{\Gamma }={\int }_{\mathrm{\Gamma }}\frac{\partial \varphi \left(Q\right)}{\partial n}\mathrm{G}(P,Q)d\mathrm{\Gamma }.$$

(26)

The boundary is discretized into $\prime N\prime$ constant elements. For each element, the values of parameter G and parameter ϕ are treated as spatially invariant and assigned the value corresponding to the mid-element node. On using the discretization, equation (26) is rewritten into a system of $\prime N\prime$ constant boundary elements given by the expression:

$$\frac{1}{2}.\varphi (P)+\sum _{j=1}^N\left({\int }_{{\mathrm{\Gamma }}_j}{\varphi }_j\frac{\partial Q}{\partial n}d\mathrm{\Gamma }\right)=\sum _{j=1}^N\left({\int }_{{\mathrm{\Gamma }}_j}{Q}_j\frac{\partial \varphi }{\partial n}d\mathrm{\Gamma }\right),$$

(27)

where, ${\mathrm{\Gamma }}_j$ is the boundary in the j ^th element. Considering ${\int }_{{\mathrm{\Gamma }}_j}\frac{\partial Q}{\partial n}d\mathrm{\Gamma }=H_{\mathrm{ij}}$ and ${\int }_{{\mathrm{\Gamma }}_j}{Q}_{j}d\mathrm{\Gamma }=G_{\mathrm{ij}}$ the equation (27) can be rewritten as:

$$-\frac{1}{2}\varphi (P)+\sum _{j=1}^N(H_{\mathrm{ij}}.{\varphi }_j)=\sum _{j=1}^N\left(G_{\mathrm{ij}}\frac{\partial \varphi }{\partial n}\right),$$

(28)

The integrals denoted by H _ij and G _ij characterize the influence functions between element i (where the fundamental solution is applied) and any other considered element j. The floating porous box remains fixed to the VLFS as depicted in Figure 1. The boundary element mesh configuration is schematically illustrated in Figure 2.

Figure 2.

Schematic representation of the boundary element mesh.

To conduct the hydroelastic analysis of the articulated VLFS integrated with a porous box, a coupled numerical approach is implemented. The Multi-domain Boundary Element Method (MDBEM) is applied to the fluid and porous structure domains, while the Finite Difference Method (FDM) is employed in the structural domain to compute the velocity potential and the deflection of the articulated VLFS. This hybrid application of MDBEM and FDM is termed the coupled MDBEM-FDM approach. By applying the boundary conditions for the articulated VLFS with a porous box to equation (27), the resulting discretized equation can be expressed as:

$$\begin{array}{c}\multicolumn{1}{c}{-\frac{1}{2}\varphi (P)+\sum _{j=1}^N(H_{\mathrm{ij}}-\mathrm{ik}{G}_{\mathrm{ij}}){\varphi }_I+\sum _{j=1}^N\left(H_{\mathrm{ij}}{\varphi }_{\mathrm{SB}}\right)}\\ \multicolumn{1}{c}{+\sum _{j=1}^N(H_{\mathrm{ij}}-\mathrm{ik}{G}_{\mathrm{ij}}){\varphi }_T+\sum _{j=1}^N(H_{\mathrm{ij}}-\frac{{\omega }^2}{g}{G}_{\mathrm{ij}}){\varphi }_{\mathrm{FS}1}}\\ +\sum _{j=1}^N{H}_{\mathrm{ij}}{\varphi }_{{\mathrm{STR}}_{1-3}}+\sum _{j=1}^N{H}_{\mathrm{ij}}{\varphi }_{F{P}_{1-n}}+i\omega \sum _{j=1}^N{G}_{\mathrm{ij}}{\eta }_{F{P}_{1-n}}\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^N{H}_{\mathrm{ij}}{\varphi }_{{\mathrm{STR}}_{4-6}}+\sum _{j=1}^N(H_{\mathrm{ij}}-\frac{{\omega }^2}{g}{G}_{\mathrm{ij}}){\varphi }_{\mathrm{FS}2}=0.}\end{array}$$

(29)

Equation (29) forms the final discretized system for the articulated VLFS integrated with porous rectangular breakwaters, as depicted in Figure 1. In this formulation, the primary unknowns are the surface deflections of the elastic plate. To solve this system, the governing Euler-Bernoulli beam equation for the elastic VLFS (equation (15)) is discretized using the Finite Difference Method, specifically employing a fourth-order central difference scheme. Applying this scheme to the elastic plate boundary condition (equation (15)) for the j ^th element yields the discretized form:

$$\begin{array}{c}\multicolumn{1}{c}{\left(\frac{{\eta }_{j+2}-4{\eta }_{j+1}+6{\eta }_j-4{\eta }_{j-1}+{\eta }_{j-2}}{{\Delta }_j^4}\right)}\\ \multicolumn{1}{c}{+\alpha \left(\frac{{\eta }_{j+1}-2{\eta }_j+{\eta }_{j-1}}{{\Delta }_j^2}\right)+\beta {\eta }_j=\chi {\varphi }_j,}\end{array}$$

(30)

where, $\alpha =-2k_z^2$ , $\beta =k_z^4+\left(\frac{\rho g-m_p{\omega }^2}{\mathrm{EI}}\right)$ , $\chi =\frac{i\rho \omega }{\mathrm{EI}}$ , $k_z=k\sin \theta$ , k is the wave number and θ is the angle of incidence. The discretized edge condition for the VLFS at $x=0,L$ are given as:

$$\frac{{\eta }_{j+1}}{{\Delta }_j^2}-(\frac{2}{{\Delta }_j^2}+{\mathrm{vk}}_z^2){\eta }_j+\frac{{\eta }_{j-1}}{{\Delta }_j^2}=0$$

(31)

$$\begin{array}{c}\multicolumn{1}{c}{\frac{1}{{\Delta }_j^3}{\eta }_{j+2}-[(\frac{2}{{\Delta }_j^3}+\frac{2-v}{{\Delta }_j}{k}_z^2)+\frac{Q_j}{\mathrm{EI}{\Delta }_j}]({\eta }_{j+1}-{\eta }_{j-1})}\\ \multicolumn{1}{c}{-\frac{1}{{\Delta }_j^3}{\eta }_{j-2}=0}\end{array}$$

(32)

The primary unknowns to be determined are the velocity potentials ϕ _j and the plate displacement η _FP . In Equation (29), the boundaries are designated as follows: ${\mathrm{\Gamma }}_{\mathrm{IR}}$ for the far-field, ${\mathrm{\Gamma }}_{\mathrm{SB}}$ for the seabed, ${\mathrm{\Gamma }}_O$ for the free surface, ${\mathrm{\Gamma }}_{\mathrm{STR}}$ for the floating structure, and ${\mathrm{\Gamma }}_{\mathrm{FS}}$ for the interface. To solve the resulting system of linear equations, matching boundary conditions are imposed to resolve the unknown velocity potentials ϕ _I , ϕ _R , and ϕ _T . The system of equation is computationally solved using the Gauss-elimination technique, enabling the determination of all unknown velocity potentials in each domain. Finally, the reflection coefficient K _r , transmission coefficient K _t , and dissipation coefficient K _d are derived by applying the law of conservation of energy. This energy balance relation satisfies the energy identity established mathematically represented by the expression:

$$K_r^2+K_t^2+K_d^2=1,$$

(33)

$$\mathrm{where}{K}_r=\left|\frac{R_0}{I_0}\right|,K_t=\left|\frac{T_0}{I_0}\right|\mathrm{and}{K}_d=\sqrt{1-K_r^2-K_t^2}.$$

(34)

The energy identity shown in equation 23(a) is derived using the Greens identity given by:

$${\int }_{\mathrm{\Gamma }}(\varphi \frac{\partial {\varphi }^{*}}{\partial n}-{\varphi }^{*}\frac{\partial \varphi }{\partial n})\mathrm{ds}=0,$$

(35)

where, ${\varphi }^{*}$ denotes the complex conjugate of ϕ, while $\frac{\partial }{\partial n}$ represents the outward drawn normal derivative to boundary Γ. The wave forces acting on the structure (Patil and Karmakar ³¹ ) are determined through the mathematical relation expressed as:

$$F=\mathrm{Re}\left[i\rho \omega {\int }_{{\mathrm{\Gamma }}_{\mathrm{SR}{R}_j}}\left\{\begin{array}{c}{n}_x\hfill \\ \multicolumn{1}{c}{n_y}\end{array}\right\}{\varphi }_{\mathrm{STR}}d\mathrm{\Gamma }\right]$$

(36)

Therefore, F can be expressed as $F=\left\{\begin{array}{c}{F}_x\hfill \\ \multicolumn{1}{c}{F_y}\end{array}\right\}$ , where n _x and n _y represent the unit normal vectors in the x-direction and y-direction, respectively, acting on ${\mathrm{\Gamma }}_{\mathrm{STR}}$ . Consequently, the horizontal wave force coefficient K _Fx on the leeside of the structure can be expressed as:

$$K_{\mathrm{Fx}}=\frac{F_x}{\rho g{I}_0{h}^2}.$$

(37)

The bending moment of the floating elastic plate is given by the relation:

$$\mathrm{BM}=\mathrm{EI}\{\frac{{\partial }^2}{\partial x^2}-\upsilon k_z^2\}\eta \left(x\right)\mathrm{on}y=0.$$

(38)

The discretized form of the bending moment ${\mathrm{BM}}^{*}=\mathrm{BM}/\mathrm{EI}$ on applying the central difference scheme is given by:

$${\mathrm{BM}}^{*}=\{\frac{({\eta }_{j+1}+{\eta }_{j-1})}{\Delta x_j^2}-[\frac{2}{\Delta x_j^2}+\upsilon k_z^2]{\eta }_j\}.$$

(39)

The shear force of the floating elastic plate is given by the relation:

$$\mathrm{SF}=\mathrm{EI}\frac{\partial }{\partial x}\{\frac{{\partial }^2}{\partial x^2}-(2-\upsilon )k_z^2\}\eta \left(x\right)\mathrm{on}y=0.$$

(40)

The discretized form of the shear force ${\mathrm{SF}}^{*}=\mathrm{SF}/\mathrm{EI}$ on applying the central difference scheme is given by:

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{SF}}^{*}=\frac{1}{2\Delta x_j^3}({\eta }_{j-2}-{\eta }_{j+2})-[\frac{1}{\Delta x_j^3}+\frac{(2-\upsilon )k_z^2}{2\Delta x_j}]}\\ \multicolumn{1}{c}{({\eta }_{j+1}+{\eta }_{j-1}).}\end{array}$$

(41)

Further, the strain in the floating elastic plate is given by the relation:

$$\tau =\frac{d}{2}\{\frac{{\partial }^2}{\partial x^2}-\upsilon k_z^2\}\eta \left(x\right)\mathrm{on}y=0.$$

(42)

The discretized form of the strain in the floating elastic plate ${\tau }^{*}=\tau /d$ on applying the central difference scheme is given by:

$${\tau }^{*}=\{\frac{({\eta }_{j+1}+{\eta }_{j-1})}{2\Delta x_j^2}-[\frac{1}{\Delta x_j^2}+\frac{\upsilon k_z^2}{2}]{\eta }_j\}.$$

(43)

In the relations (38) to (43), η _j is the surface deflection of the j ^th element obtained by solving the system of equations, $\Delta x_j$ is the element length under consideration.

Numerical results and discussion

This section comprehensively details the numerical investigation into the hydroelastic performance of articulated VLFS integrated with porous rectangular breakwaters, considering a variable number of articulations. The study systematically evaluates how various geometrical and structural parameters influence the integrated VLFS-breakwater system’s effectiveness. The accuracy of the computational model, which utilizes the combined multi-domain boundary element method and finite difference scheme (MDBEM–FDS), is rigorously validated against published results.

Table 1 details the structural and geometric parameters applied in the hydroelastic analysis, specifying the numerical values used as defaults throughout this study. This standardized set of parameters ensures a consistent framework for evaluating the critical interactions between wave forces, breakwater porosity, the number of VLFS articulations, and VLFS flexibility. The subsequent sections will provide a comprehensive analysis and discussion on the effects of varying the geometric properties of the integrated VLFS and porous barrier system.

Table 1.

Structural and geometrical parameters of integrated VLFS-breakwater system.

Structure	Parameters		Values
Very large floating structure	Thickness	d/h	0.005–0.050
	Length	L/h	5.0–10.0
	Elasticity modulus	E	5–10 GPa
	Vertical linear spring stiffness	k 33	0–109 N/m
	Rotational spring stiffness	k 55	0–109 Nm
Porous breakwater	Width	b/h	0.5–2.0
	Height	$h_1/h$	0.15–0.50
	Porosity	ε	0.1–0.4
Common parameters	Mooring stiffness	q	0–109 N/m
	Angle of incidence	θ	0°–90°

Convergence of the MDBEM approach

The convergence of the Multi-Domain Boundary Element Method (MDBEM) approach for wave interaction with the VLFS integrated with porous rectangular breakwaters under varying number of VLFS articulations is documented in Table 2. The study establishes the convergence of K _r and K _t up to four decimal places at constant element lengths $\Delta x=0.1$ , $\Delta x=0.05$ , and $\Delta x=0.01$ .

Table 2.

Convergence of K _r and K _t with the incremental boundary elements.

${\mathrm{\Gamma }}_{\mathrm{SB}}$	${\mathrm{\Gamma }}_{\mathrm{OUT}}$	${\mathrm{\Gamma }}_{F{S}_1}$	${\mathrm{\Gamma }}_{\mathrm{ST}{R}_1}$	${\mathrm{\Gamma }}_{\mathrm{ST}{R}_2}$	${\mathrm{\Gamma }}_{\mathrm{ST}{R}_3}$	${\mathrm{\Gamma }}_{F{S}_2}$	${\mathrm{\Gamma }}_{\mathrm{IN}}$	Δx	N	K r	K t
120	10	10	1	100	1	10	10	0.1	262	0.08944	0.96318
240	20	20	2	200	2	20	20	0.05	524	0.09586	0.99423
1200	100	100	10	1000	10	100	100	0.01	2620	0.09513	0.99747

Further, Table 3 shows computational durations required for result generation across these element lengths. The observations indicate that convergence for K _r and K _t occurs at element lengths $\Delta x=0.05$ and $\Delta x=0.01$ , though computational time significantly increases for element length $\Delta x=0.01$ . Consequently, the constant element length $\Delta x=0.05$ is selected for efficient structural parameter analysis, corresponding to the standard element discretization size = 1/20th per meter. Thus, it is advisable to consider the element length of $\Delta x=0.05$ , as this choice saves computational effort without compromising with the results. In the present study, at element lengths of $\Delta x=0.05$ and $\Delta x=0.01$ shows same results without any deviation.

Table 3.

Computational time for various element lengths.

Element length	0.1	0.05	0.01
Total elements	524	924	4604
Computation time	41 min	2 h–33 min	27 h

Validation of the numerical model

The numerical model implemented using the MDBEM approach has been validated against several established studies, demonstrating strong agreement. This includes validation with the work of Mei and Black for a rectangular rigid obstacle, Azm and Gesraha ³⁸ for a floating porous structure, and Andrianov and Hermans ³⁹ for a floating elastic VLFS. As illustrated in Figure 3(a), the model is validated for a fixed pontoon based on the results of Azm and Gesraha. ³⁸ This specific validation is performed for a permeable structure under conditions of relative water depthd/h=0.25 and relative width B/h=2 using the present MDBEM methodology.

Figure 3.

Validation of (a) K _r and K _t for a rectangular floating porous breakwater with Azm and Gesraha ³⁸ and (b) K _r for a rectangular rigid floating obstacle with Mei and Black. ⁴⁰

The results demonstrate that the wave transmission coefficient reaches its maximum, while the reflection coefficient is minimized, at normal angles of wave incidence. The analysis also observes variations in the reflection coefficient with increasing kh. The numerical results from the MDBEM approach show excellent agreement with the findings reported by Azm and Gesraha. ³⁸ Figure 3(b) presents the numerical results for a rigid floating obstacle with a relative depth ofH/h=0.2 and a width of l/h=1.0 at a normal angle of incidence. These findings are validated against the results of Mei and Black. ⁴⁰ The reflection coefficient obtained from the MDBEM approach aligns acceptably with the published data, confirming the reliability of the present numerical model.

In Figure 4, the validation of the interaction of waves with hinged VLFS is performed as in Kohout and Meylan. ⁴¹ The validation considers the variation in angle of incidence from $\theta ~0^0-{50}^0$ , and a relative depth of H/h=1. The resulting reflection coefficient displays minor variations in both K _r and K _t for varying θ which remains within an acceptable range when contrasted with the results of Kohout and Meylan. ⁴¹ These findings reinforce the efficacy of the MDBEM-FDC approach in modeling wave interactions accurately.

Figure 4.

Validation of reflection K _r and transmission coefficients K _t for articulated VLFS with Kohout and Meylan. ⁴¹

Effect of single articulation

The hydroelastic analysis is performed for a porous rectangular breakwater integrated with single articulated VLFS, varying structural and geometric parameters of the rectangular breakwater. A comprehensive study of the surface displacement is presented for all configurations with all parametric modifications, considering oblique incident waves.

Reflection and transmission coefficients

Figure 5(a) and (b) represents the reflection and transmission coefficients against wave number kh for the single articulated VLFS integrated with porous rectangular breakwaters on either side for varying width of the breakwater b/h and modulus of elasticity E for VLFS . Figure 5(a) shows that the transmission K _t gradually reduces with an increase in b/h. It is also observed that the wave transmission is lower than the reflection coefficients which may be due to the loss of turbulence of the waves due to the elastic joints. On the other hand, the reflection coefficients K _r are significantly higher in the shallow and intermediate water depths due to the pressure difference created due to the porous breakwaters. Figure 5(b), clearly shows that the resonance is not carried forward across the entire structure but is localized in certain regions. The elastic joints play a crucial role in damping the secondary forces. In addition, the wave transmission is seen to be lower in deeper water depths. At kh≈ 2.5, K _t drops 45% when b/h increases from 0.2 to 0.4, proving articulations synergize with breakwaters for broadband wave attenuation. Further, the joint rotational stiffness should be 10%–20% of module flexural rigidity for peak K _t reduction.

Figure 5.

K _r and K _t versus kh for different (a) width b/h of the porous breakwater and (b) modulus of elasticity E of the VLFS for single articulation.

Surface deflections

Figure 6(a) and (b) shows the surface deflection η of the VLFS along the length of the plate for varying porosity of the breakwater ε, and elastic modulus E of the VLFS considering single articulation. Figure 6(a) shows that the surface deflections increase with the increase in porosity of the breakwaters which may be due to the availability of a larger surface area for interparticle movement, thereby increasing the flow. It is also seen that after the articulation, the surface deflections decrease for all variations, also helping in reducing the overall maintenance cost. Figure 6(b) shows that the deflections decrease with the increase in the elastic modulus of the VLFS due to the increase in permanent internal rigidity offered by the structure. Hence, structures with larger elastic modulus are preferred.

Figure 6.

Surface deflections η versus x/L for different (a) porosity ε of the breakwater and (b) modulus of elasticity E of the VLFS for single articulation.

Shear force on the floating plate

Figure 7(a) and (b) shows the shear force coefficients of the VLFS along the length of the plate for varying width b/h of the porous breakwater and elastic modulus E of the VLFS with single articulation. Figure 7(a) illustrates that an increase in the relative width b/h results in a global reduction of shear forces ${\mathrm{SF}}^{*}$ , especially at midspans. The peak shear force near the articulation significantly decreases for the case of a larger width b/h. The wider breakwaters facilitate greater wave energy dissipation through porous flow, which in turn reduces wave transmission to the VLFS.

Figure 7.

Shear force coefficients ${\mathrm{SF}}^{*}$ along the length of the plate x/L for different (a) width b/h of porous breakwater and (b) elastic modulus E of VLFS with single articulation.

The decrease in hydrodynamic pressure on the structure lowers bending and shear loads, particularly at the articulation point, which is a high-stress area. Additionally, a larger width b/h enables the design of thinner VLFS by reducing peak shear forces. However, excessive width may lead to increased construction costs without corresponding benefits. In Figure 7(b), the high elastic modulus E leads to sharper spikes in the shear force ${\mathrm{SF}}^{*}$ near the articulation point and supports. The high elastic modulus tend to concentrate shear forces at discontinuities, whereas flexible VLFS dissipate energy through deformation. A lower elastic modulus E can avoid wave-frequency synchronization, thereby reducing dynamic amplification.

Bending moment on the floating plate

Figure 8(a) and (b) shows the bending moment of the VLFS along the length of the plate for varying height $h_1/h$ of the porous breakwater and thickness d/h of the VLFS considering single articulation. In Figure 8(a), for lower relative height of porous breakwater $h_1/h$ there is a pronounced peak, at the articulation point, which reaches around 2.0–2.5 times those at the ends. This results in a flatter distribution and ∼30% reduction in end moments. Taller breakwaters disrupt the wave pressure distribution over depth, blocking 60%–80% of wave energy. The increased height suppresses flow separation vortices at the edges of the breakwater, thereby minimizing the turbulent fluctuations that can amplify moments. The midspan hinge area concentrates bending forces; however, taller breakwaters protect this zone by absorbing energy upstream. The relative heights >0.35 increase drag forces without a proportional reduction in moments. In Figure 8(b), a broad, smooth moment distribution is observed, with the end moments approaching zero. The rigidity of the system prevents the redistribution of loads, concentrating moments at the articulation point and demonstrating a deviation. Thin plates that conform the surface displacements is due to the reduced pressure gradients, while thick plates are subjected to constructive wave interference. To limit bending spikes while managing deflection, it is advisable to use a thickness d/h= 0.01 near the articulation point. The breakwater height plays a more significant role in this scenario, and it is recommended to optimize the $h_1/h$ before addressing the thickness of the plates.

Figure 8.

Bending moment ${\mathrm{BM}}^{*}$ versus x/L for different (a) relative height $h_1/h$ of the porous breakwater and (b) thickness d/h of the VLFS with single articulation.

Strain in the floating elastic plate

Figure 9(a) and (b) shows the strain in the floating VLFS along the length of the plate for varying the porosity ε of the breakwater and stiffness q of the mooring cables connected to the VLFS with a single articulation. In Figure 9(a), lower-porosity breakwaters function as near-solid barriers, reflecting 70%–80% of wave energy. This reflection amplifies pressure gradients, resulting in higher strain at discontinuities. On the other hand, higher porosity increases wave transmission, which restores pressure loads on VLFS. An optimal porosity range of 20%–30% maximizes viscous dissipation in the pores. This allows flow separation vortices to convert a small portion of kinetic energy, reducing the transmitted wave height by 40%–60%, which helps lower pressure-induced strain. In Figure 9(b), stiff moorings exhibit a 25%–30% higher peak strain at articulation compared to soft moorings. High stiffness in the moorings aligns the natural frequencies of the VLFS with the wave periods that resonate for strain amplification. However, overly soft moorings risk collisions, while stiff moorings can accelerate fatigue at the hinges. Porosity influences external energy input, while mooring stiffness governs structural kinematics. Strain tends to concentrate at articulation points due to discontinuities in bending stiffness, which are optimized by balancing wave dissipation.

Figure 9.

Strain in the floating elastic plate ${\tau }^{*}$ along the length of the plate x/L for different (a) porosity ε of the breakwater and (b) stiffness q of the mooring cables connected to the VLFS with single articulation.

Wave force on the breakwater

Figure 10(a) and (b) represents the wave force coefficients against wave number kh for a VLFS integrated with porous rectangular breakwaters for varying relative heights $h_1/h$ of the porous breakwater and relative lengths L/h of the VLFS for single articulation. Figure 10(a) illustrates that shorter breakwaters exhibit higher K _FX values across all kh values, with a sharp resonance peak occurring near $\mathrm{kh}\approx 1.5$ . In contrast, taller breakwaters can reduce K _FX by 30%–50%, eliminating the resonance peak and flattening the response curve. Shorter breakwaters expose the VLFS to wave action, allowing waves to propagate underneath and amplify hydrodynamic pressures, which in turn excites resonant vibrations near $\mathrm{kh}\approx 1.5$ . Taller breakwaters extend deeper into the water column, blocking wave energy transmission and enhancing viscous dissipation. This disruption of wave-structure resonance leads to a reduction in force transmission. Figure 10(b) demonstrates that the wave force distribution becomes broader and flatter for longer structures. Shorter VLFS have higher natural frequencies, which synchronize with common storm wave resonances, resulting in force amplification. Conversely, longer VLFS have lower natural frequencies, causing them to detune from the dominant wave periods. Their extended length allows for a more even distribution of wave loads, thereby reducing localized pressures and avoiding sharp resonance. It is recommended to use the relative length of $L/h\ge 8$ to shift the resonance away from high-risk wave numbers and reduce peak forces by 25%. The breakwater height regulates wave energy transmission, while VLFS length influences resonant behavior. On optimizing both factors, it is possible to transform sharp increase in the wave force peaks associated with minimal designs into a more subdued and predictable response. This adjustment enables safer VLFS operations in a variety of sea states.

Figure 10.

Wave force coefficient K _FX versus kh for different (a) relative height $h_1/h$ of the porous breakwater and (b) relative length L/h of the VLFS for single articulation.

Effect of two articulations

The hydroelastic analysis is performed for a porous rectangular breakwater integrated with VLFS with two articulations, varying structural and geometric parameters of the rectangular breakwater. A comprehensive comparison of the surface displacement is presented for all configurations with all parametric modifications, considering oblique incident waves.

Reflection and transmission coefficients

Figure 11(a) and (b) represents the reflection and transmission coefficients against wave number kh for the VLFS integrated with porous rectangular breakwaters on either side for varying widths of the porous breakwater b/h and modulus of elasticity E of the VLFS in the case of two articulations. Figure 11(a) demonstrates that narrow breakwaters produce low reflection ( $K_r<0.3$ ) and high transmission ( $K_t>0.7$ ), indicating ineffective wave protection. The optimal performance of the narrow width of the breakwater is observed during long period waves. Narrow breakwaters permit waves to pass through with minimal energy loss, whereas wider breakwaters increase the path of wave energy dissipation through porosity of the structure. Turbulent friction converts wave kinetic energy into heat, thus reducing wave transmission. The wave reflection increases because wider barriers redirect more wave energy towards seaward region. It is recommended to avoid relative width $b/h<1.0$ in high-energy locations, as a transmission coefficient >0.6 can compromise the stability of VLFS.

Figure 11.

K _r and K _t versus kh for different (a) relative width b/h of the porous breakwater and (b) modulus of elasticity E of the VLFS with two articulations.

Figure 11(b) reveals that flexible VLFS slightly increases wave reflection while decreasing wave transmission for $\mathrm{kh}<2.5$ . Flexible VLFS deform in response to waves, dissipating energy through elastic hysteresis and enhance wave reflection due to structural compliance. In contrast, stiff VLFS resist deformation, allowing more wave energy to pass beneath the structure, particularly for long-period waves. The width of the breakwater is the primary factor in wave energy dissipation, while the stiffness of the VLFS fine-tunes the balance between the wave reflection and transmission. Optimizing both can lead to ‘climate-adaptive’ VLFS systems, which can switch between wave-reflecting modes in calm seas and wave-blocking modes during long-period waves. Breakwaters effectively absorb high-frequency waves (kh > 2), while flexible VLFS dampen long-period swells (kh < 1). This dual mechanism extends the service life of VLFS in regions vulnerable to climate change by reducing cyclic wave loads on the joints and articulations.

Surface deflection

Figure 12(a) and (b) shows the surface deflection of the VLFS along the length of the plate for varying height of the rectangular breakwater $h_1/h$ and stiffness q of the mooring cables connected to the VLFS in the case of two articulations. Figure 12(a) illustrates that shorter breakwaters result in significant deflections, particularly peak at the articulations and mid-segments. In contrast, taller breakwaters ( $h_1/h>0.4$ ) reduces peak deflection by 40%–50% and flatten the profile, especially at the articulations. Shorter breakwaters allow waves to pass underneath, creating hydrodynamic pressures that cause bending at the articulations. Taller breakwaters block the transmission of wave energy, thereby reducing pressure loads. It is recommended to adopt $h_1/h>0.4$ to limit the maximum deflection ${\eta }_{max}$ for operational stability.

Figure 12.

Surface deflections $\eta (x)$ along the length of the plate x/L for different (a) relative height $h_1/h$ of the rectangular breakwater and (b) stiffness q of the mooring cables connected to the VLFS with two articulations.

In Figure 12(b), soft moorings exhibit moderate deflection but significant uplift at the ends, while stiff moorings increase deflection at the articulations by 30% and suppress end movement. Soft moorings permit low-frequency surge, which tilts the VLFS and lifts the ends. Stiff moorings transfer wave impact forces directly to the articulations, which increases local bending. Intermediate stiffness allows for controlled drift, dissipating energy without resonant amplification. Overall, the system functions as a dual-stage isolator. The breakwaters absorb short-period waves, while the optimized moorings dampen low-frequency motions. This combination reduces strains at the articulations by 40%, extending maintenance intervals and ensuring service continuity in extreme climatic conditions.

Shear force on the floating elastic plate

Figure 13(a) and (b) presents the shear force of the VLFS along the length of the plate for varying relative width b/h of the porous breakwater and elastic modulus E of the VLFS in the case of two articulations. Figure 13(a) illustrates that peak shear forces at the articulation decrease by ∼30%–40% as the relative width b/h increases. Additionally, an increase in b/h results in a reduction in ${\mathrm{SF}}^{*}$ , with the most significant attenuation occurring at articulations and ends of the VLFS. The shear distribution becomes smoother with wider breakwaters, which helps to reduce stress concentrations. As hydrodynamic pressure on VLFS decreases, bending moments are also reduced, leading to lower shear forces at critical articulations. In Figure 13(b), it can be seen that stiff structures display sharp spikes in shear at the articulations, where ${\mathrm{SF}}^{*}$ peaks are two times higher than in normal scenarios. A high modulus of elasticity E restricts the structure’s ability to conform the wave surface, which amplifies pressure differentials and results in increased shear at discontinuities. In contrast, flexible structures distribute shear more evenly, reducing peak magnitudes by 25%–50%. Moreover, a higher E may align with wave frequencies, which can increase dynamic shear. With two articulations, the redistribution of shear becomes more complex compared to single articulation. The hinges located at one-third and two-thirds along the structure divide it into three segments, concentrating shear at these points. The breakwaters are most effective in mitigating this concentration at higher relative width b/h, while the flexibility of the structure helps to smooth out shear distributions.

Figure 13.

Shear forces ${\mathrm{SF}}^{*}$ versus x/L for different (a) relative width b/h of the porous breakwater and (b) elastic modulus E of the VLFS with two articulations.

Bending moment on the floating elastic plate

Figure 14(a) and (b) shows the bending moment coefficients of the VLFS along the length of the plate for varying widths b/h of porous breakwater and stiffness q of the mooring cables connected to the VLFS in the case of two articulations. Figure 14(a) illustrates that wider breakwaters increase the path length for wave propagation through porous media, which enhances viscous dissipation. The dissipated wave energy helps minimize pressure differentials across the articulations, directly reducing curvature-induced bending and suppressing vortex shedding around the edges of the breakwater. As a result, turbulent pressure fluctuations are minimized. On combining with the porosity of 20%–30% from prior analysis, this design yields maximum energy dissipation. The relative width within $1.5<b/h<2.0$ can reduce peak bending by 35%–40% while maintaining moderate material costs. Figure 14(b) shows that soft moorings exhibit moderate ${\mathrm{BM}}^{*}$ peaks at articulation points, resulting in the lowest overall bending moments. The soft mooring increases low-frequency oscillations, and also raises quasi-static bending at the ends. Nevertheless, this design allows the VLFS to drift with the waves, thereby reducing the wave-structure relative velocity and lowering the dynamic pressure. In the case of stiff moorings, it is recommended to reinforce articulation at x/L= 3.75 with elastomeric bearings. Overall, the soft moorings may lead to excessive drift, while stiff moorings can increase structural fatigue. The width of the breakwater governs the reduction of wave loads, whereas mooring stiffness controls the kinematic constraints.

Figure 14.

Bending moment ${\mathrm{BM}}^{*}$ along the length of the plate x/L for different (a) relative width b/h of the porous breakwater and (b) stiffness q of the mooring cables connected to the VLFS with two articulations.

Strain in the floating elastic plate

Figure 15(a) and (b) shows the strain in the floating VLFS along the length of the plate for varying different breakwater porosity ε and mooring line stiffness q for VLFS connected with two articulations. Figure 15(a) illustrates strain rebound at higher porosities. The increased wave transmission restores pressure loads, while larger pores reduce flow resistance, which weakens energy dissipation and leads to partial strain recovery. Articulations create bending stiffness discontinuities, resulting in localized curvature under load. Reflected waves (with low strain, ${\tau }^{*}$ ) amplify pressure differentials. The peak strain at the mid-articulation is 2.0–2.2 times higher than ε. Employing graded porosity to balance reflection and dissipation will be more economical in the long run. Figure 15(b) indicates that restricted horizontal motion increases the relative wave-structure velocity and amplifies wave impact forces. A high q synchronizes the natural frequencies of a VLFS with wave periods, causing resonant strain amplification at the articulations. Stiff moorings, which approximate clamped ends, induce high curvature, resulting in strain concentration at the ends that propagates to the articulations. The strain peaks at articulations arise from the interaction between bending stiffness discontinuities and wave-induced pressures. Breakwater porosity controls wave load intensity by regulating energy reflection and dissipation. Mooring stiffness influences structural kinematics; stiff constraints amplify dynamic loads, while excessive strain can excite low-frequency oscillations. Optimal pairing of these elements can reduce peak strain by 40%–50%.

Figure 15.

Strain ${\tau }^{*}$ versus x/L for (a) different breakwater porosity ε and (b) mooring line stiffness q for VLFS connected with two articulations.

Wave force on the breakwater

Figure 16(a) and (b) represents the wave force coefficients against wave number kh for a VLFS integrated with porous rectangular breakwaters on either side for varying porosity ε of the breakwater and elastic modulus E of the VLFS in the case of two articulations. Figure 16(a) demonstrates that breakwaters with lower porosity reflect ∼60% of wave energy with the increase in porosity, leading to the creation of standing waves that intensify pressures on the VLFS at critical wave numbers kh, which indicates wave-structure resonance. Moderate porosity levels (20%–30%) optimize viscous dissipation within the pores, converting wave energy and disrupting the build-up of resonant energy. In contrast, high porosity allows for partial wave transmission, which can restore pressure loads at elevated kh values. Figure 16(b) illustrates that a stiff VLFS displays narrow resonance peak at $\mathrm{kh}\approx 1.8$ and increases its structural natural frequencies, aligning with typical long-wave conditions. Lower elastic modulus E contributes to enhanced structural compliance, enabling the VLFS to deform in response to wave actions and dissipate energy through elastic hysteresis. This mechanism helps to detune resonance effects and distribute loads more evenly. In regions where kh values are predominantly between 1.5 and 2.0 (such as long-wave condition), it is advisable to prioritize porosity levels >0.25 or utilize broad wave spectra (in open ocean) and flexible materials with an $E\le 7\mathrm{GPa}$ to achieve a flatter force response. Breakwater porosity plays a crucial role in governing wave energy reflection and dissipation, while the stiffness of the VLFS controls its resonant behavior. By optimizing both factors, it is possible to transform sharp and hazardous force peaks into a damped response, thereby improving the safety of VLFS operations across a range of sea states.

Figure 16.

Wave force coefficients K _FX versus kh for different (a) porosity ε of the breakwater and (b) elastic modulus E of the VLFS with two articulations.

Effect of three articulations

The hydroelastic analysis is performed for a porous rectangular breakwater integrated with VLFS with three articulations, varying structural and geometric parameters of the rectangular breakwater. A comprehensive comparison of the surface displacement is presented for all configurations with all parametric modifications, considering oblique incident waves.

Reflection and transmission coefficients

Figure 17(a) and (b) shows the reflection and transmission coefficients against wave number kh for VLFS integrated with porous rectangular breakwaters on either side for width b/h of the breakwater and elastic modulus E of the VLFS with three articulations. Figure 17(a) illustrates that narrow breakwaters allow waves to pass through with minimal energy dissipation. In contrast, wider breakwaters increase wave reflection by 60%–80% ( $K_r\approx 0.8$ ) while reducing wave transmission by 40%–50% ( $K_t\approx 0.3$ ) for long-period waves. The increased width of these breakwaters extends the path of wave energy dissipation through porous materials, enhancing turbulent friction and converting wave energy. This process reduces wave transmission and increases reflection due to the greater effectiveness of the breakwater. Figure 17(b) demonstrates that stiff VLFS can increase wave transmission by 10%–20% for low kh values while matching the performance of flexible VLFS for short waves (kh > 2). Flexible VLFS deform in response to waves, dissipating energy through elastic hysteresis and enhancing reflection due to their structural compliance. On the other hand, stiff VLFS resist deformation, allowing more wave energy to pass beneath the structure, particularly for long-period swells, where flexibility plays a critical role. In summary, breakwater width primarily controls wave energy dissipation, while the stiffness of VLFS fine-tunes reflection. Together, the breakwater width and modulus of elasticity form a dynamic wave barrier that adapts to varying sea states: reflecting long swells through flexibility and dissipating waves through width adjustments.

Figure 17.

K _r and K _t versus kh for different (a) relative width b/h of the breakwater and (b) elastic modulus E of the VLFS with three articulations.

Surface deflection

Figure 18(a) and (b) shows the surface deflection of the VLFS along the length of the plate for varying height $h_1/h$ of the rectangular breakwater and porosity of the breakwater ε integrated to a VLFS with three articulations. In Figure 18(a), the mid-span articulation experiences a deflection, that is, 25% greater than that of the end hinges, a result of symmetric wave focusing. The deflection patterns indicate a coupled articulated response; specifically, reduced deformation at one hinge leads to amplified motion at adjacent articulation when the ratio $h_1/h<0.4$ . The central hinge serves as a convergence point for wave energy in symmetric sea conditions. Taller breakwaters effectively suppress shorter-period waves, but they struggle with long-period swells ( $\lambda >2L$ ) that can penetrate beneath the central zone. Figure 18(b) illustrates that three articulations create coupled resonance cavities between the articulated edge. Low porosity excites these cavity modes, while high porosity allows for wave coherence across the segments. The optimal porosity level disrupts phase matching across zones separated by hinges. Increased porosity raises the mid-segment deflection by 15% due to wave channeling between breakwaters. In the case of three-articulated VLFS, controlling deflection requires spatially tailored breakwaters that address hinge-induced resonance. This solution transforms vulnerable mid-span articulations from being wave amplifiers into energy dissipators, facilitating the development of mega-scale floating cities in typhoon-prone regions.

Figure 18.

Surface deflections $\eta (x)$ along the length of the plate x/L for different (a) relative height $h_1/h$ of the rectangular breakwater and (b) porosity ε of the breakwater integrated to a VLFS with three articulations.

Shear force on the floating elastic plate

Figure 19(a) and (b) shows the shear force of the VLFS along the length of the plate for varying porosity of the breakwater ε and elastic modulus E of the VLFS with three articulations. Figure 19(a) illustrates the highest ${\mathrm{SF}}^{*}$ , which displays sharp peaks at the articulations. The reduction of 15%–25% in peak ${\mathrm{SF}}^{*}$ is observed at the case of three articulations when the porosity is lower. An optimal porosity range of 20%–30% effectively minimizes hinge shear forces. Moderate porosity, defined as <30%, facilitates optimal wave energy dissipation through porous flow, thus reducing the transmitted wave loads. In the case of lower value of porosity ε, denser breakwaters become necessary, leading to increased costs; however, lower ε also saturates the effects of wave attenuation. The presence of pores generates turbulent vortices, which dissipate 40%–60% of wave energy. The reduction in energy allows partially transmitted waves to lessen pressure gradients on the VLFS. Consequently, shear force decreases as energy is redirected into fluid friction instead of being applied as structural loading. This breaks wave coherence, preventing synchronized loading across articulations. However, higher porosity permits over 70% wave transmission, thus restoring pressure loads on the VLFS. Figure 19(b) indicates that greater rigidity inhibits conformance to wave profiles, resulting in high curvature at the articulations. Over-damping can cause peaks in shear forces due to unbalanced dynamic reactions at the articulations, which arise from inertial dominance. Lower material stiffness E reduces mid-hinge shear by 30%–40%, which is crucial for resilience against earthquakes and storms. Conversely, high E minimizes deflection but amplifies hydrodynamic vibrations.

Figure 19.

Shear force ${\mathrm{SF}}^{*}$ along the length of the plate x/L for different (a) porosity ε of the breakwater and (b) elastic modulus E of the VLFS with three articulations.

Bending moment on the floating elastic plate

Figure 20(a) and (b) shows the bending moment coefficients of the VLFS along the length of the plate for varying relative widths b/h of porous breakwater and stiffness q of the mooring cables connected to the VLFS with three articulations. Figure 20(a) illustrates a 40%–50% reduction in peak bending at the articulations, with the most significant effect occurring at the mid-articulation x/L= 5.0. Moments at the midpoints of the segments decrease by 30%. Taller breakwaters disrupt pressure distribution throughout the water depth, blocking 70% of incident wave energy. The reduction in hydrodynamic pressure gradients on VLFS lowers the curvature-driven bending at the articulated ends. Full-depth coverage helps to minimize flow separation, thereby preventing turbulent pressure fluctuations that can amplify moments. The highest bending occurs at x/L=5.0 due to symmetric wave loading; taller breakwaters effectively shield this zone. Figure 20(b) demonstrates that soft moorings show moderate bending ${\mathrm{BM}}^{*}$ at the articulations but lead to elevated moments at the ends and midpoints of the segments. This situation can excite low-frequency surge and increase quasi-static bending at the ends and mid-segments. An intermediate q can detune the natural frequencies of the VLFS from the dominant wave periods. To achieve a 50% reduction in peak bending at the mid-articulation, it is effective to have $h_1/h>0.4$ with $q>{10}^7$ . Implementing semi-rigid connectors can help to mitigate the end moments caused by moorings. The height of the breakwater plays a crucial role in wave load mitigation, while the stiffness of the mooring controls the kinematic constraints. Ultimately, the mid-articulation is the primary area of concern for bending response and should be prioritized for protection.

Figure 20.

Bending moment ${\mathrm{BM}}^{*}$ along the length of the plate x/L for different (a) relative height $h_1/h$ of porous breakwater and (b) stiffness q of the mooring cables connected to the VLFS with three articulations.

Strain in the floating plate

Figure 21(a) and (b) shows the strain in the floating VLFS along the length of the plate for varying relative widths b/h of the porous breakwater and relative thickness d/h of the VLFS with three articulations. Figure 21(a) illustrates that wider breakwaters increase the flow path length for waves passing through porous media, which enhances wave dissipation. The dissipation of 60%–80% of wave energy is noted when the b/h is 2.0, and it reduces the transmitted wave height by 40%–60%. Additionally, wider breakwaters minimize flow separation vortices at the edges, leading to dampened turbulent pressure fluctuations. In the case of b/h within $1.5<b/h<2.0$ , mid-articulation strain is reduced by 40%–50%, which is crucial for extending the fatigue life of the structure. Lower wave transmission decreases the hydrodynamic pressure differentials on the VLFS, thereby reducing the curvature that induces strain at the articulations. Figure 21(b) shows that d/h distributes wave loads over longer segments, which minimizes strain concentration at these articulations. The strain is noted to be 60% lower at d/h= 0.01 compared to d/h= 0.005, allowing for lighter superstructures and eliminating the need for hinge reinforcement. In the case of three-articulated VLFS, it is beneficial to prioritize breakwater width ( $b/h=1.5-2.0$ ) to reduce wave loads, while also selecting VLFS thickness ( $d/h=0.02-0.03$ ) to minimize strain amplification. This combined approach transforms the critical mid-articulation from a vulnerability into a manageable component, enabling the design of safer, lighter, and more cost-effective floating structures.

Figure 21.

Strain coefficients ${\tau }^{*}$ along the length of the plate x/L for different (a) relative width b/h of the porous breakwater and (b) relative thickness d/h of the VLFS with three articulations.

Wave force on the breakwater

Figure 22(a) and (b) represents the wave force coefficients against wave number kh for a VLFS integrated with porous rectangular breakwaters on either side for varying porosity of the breakwater ε and relative width b/h of the breakwater connected to the VLFS with three articulations. Figure 22(a) illustrates that higher porosity leads to a slight rebound in force but results in lower peak forces and increased wave transmission. This effect partially restores wave forces for short waves ( $\mathrm{kh}>2.5$ ). In contrast, moderate porosity reduces peak forces by 25%–35% and flattens the wave force curve, shifting the resonance. Figure 22(b) demonstrates that narrow breakwaters offer insufficient length for energy dissipation, allowing waves to penetrate and trigger structural resonance. The wider breakwaters, on the other hand, extend the energy dissipation zone. The enhanced length increases turbulent friction and wave decay as the fluid moves through the porous media, disrupting coherent wave phases. The porosity of the breakwater directly influences energy conversion, while its width determines dissipation efficiency. Together, these factors transform dangerous resonant peaks into a damped and predictable force response, which is crucial for safer operations of VLFS in extreme sea conditions. The interplay between porosity and width will help in optimizing both parameters yields better and more cost-effective results than relying on extreme values. The dual approach extends the lifespan of structures by reducing cyclic fatigue at articulations, a vital consideration for climate-resilient floating infrastructure.

Figure 22.

Wave force coefficients K _FX versus kh for different (a) porosity ε of the breakwater and (b) relative width b/h of the breakwater connected to the VLFS with three articulations.

Comparative study of articulated floating elastic plate integrated with breakwater

A comparative study is presented to determine the effects of the number of articulations on the hydroelastic performance of the VLFS-breakwater system. A comparison between a single, two and three articulations are presented for better understanding on the effect of articulations. The variations of vertical spring stiffness and rotational stiffness are also presented to determine the influence of both springs on the hydroelastic and hydrodynamic performance of the structure.

Variation of the number of articulations

Figure 23(a) to (f) illustrates the comparative variations of reflection coefficients, transmission coefficients, shear forces, bending moments, strain, and wave force coefficients for one, two, and three articulations. Figure 23(a) and (b) demonstrate the effect of the number of articulations on the reflection and transmission coefficients. It is observed that both wave reflection and transmission decrease as the number of articulations increases. This phenomenon is attributed to the creation of additional energy dissipation zones near the articulation hinges. The impacts are particularly significant in shallow and intermediate waters, but they remain constant in deeper waters, likely due to the wave-particle motions beneath the free surface.

Figure 23.

Comparison of the performance of different articulations for (a) reflection coefficient K _r , (b) transmission coefficients K _t , (c) shear force coefficients $S{F}^{*}$ , (d) bending moment coefficients $B{M}^{*}$ , (e) strain coefficients ${\tau }^{*}$ , and (f) wave force coefficients K _FX .

Figure 23(c) to (e) reveal that with the addition of more hinges, the VLFS has shorter, more segments. As a result, the forces from waves striking the first module are not entirely transmitted to the second and third modules. Each module reacts more independently to the localized wave conditions, which prevents the cumulative buildup of shear forces that occurs in a continuous structure. The hinges introduce zero-moment connections into the system. These connections, by definition, do not allow bending moments to be transferred from one side to the other (resulting in BM* = 0 at the hinge). Consequently, the bending moment in the structure must drop to zero at each articulation point. This fundamentally alters the structural response, changing it from a continuous beam to a simply-supported or free-ended beam system for each module. This configuration inherently results in lower maximum bending moments for a given load compared to a continuous system of the same total length. Further, the wave energy is dissipated through controlled rotation at the hinges rather than being stored as elastic bending energy throughout the structure. A more articulated structure is inherently more compliant; it flexes and moves in tandem with the wave surface profile instead of resisting. This compliance is a key factor in reducing internal loads, such as shear and moment, since the structure does not need to generate large internal forces to maintain its shape. The hinges are specifically designed to serve as points of rotation. By controlling where the structure flexes, the design engineers can reinforce those specific points while lightening the rest of the structure, resulting in a more efficient and safer design. Figure 23(f) illustrates that the wave forces are slightly higher when breakwaters are integrated with a single articulated VLFS compared to those with two or three articulations. The increase in wave forces is again attributed to the dissipation of energy at the articulations. However, the effect is minimal since the breakwater encounters the waves first in all articulation scenarios.

Variation of k₃₃ and k₅₅

Figure 24(a) to (f) shows the comparative variations of vertical spring stiffness k₃₃ and rotational spring stiffness k₅₅ for the reflection coefficients, transmission coefficients, shear forces, bending moments, strain, and wave force coefficients. Figure 24(a) and (b) illustrates the reflection and transmission coefficients for varying k₃₃ and k₅₅. The analysis indicates that the k₃₃ plays a more significant role in determining the hydroelastic responses compared to k₅₅. The vertical stiffness has a greater influence on rotational stiffness due to the large surface area of the VLFS. As the stiffness of the springs increases, the overall setup behaves more rigidly. Although the increased rigidity of the joints causes the structure to behave like a rigid body, the material properties and residual elasticity prevent it from being entirely rigid. Consequently, as the vertical spring stiffness increases, the wave trapping effect diminishes, which is evident in both the reflection and transmission trends.

Figure 24.

Variation of the vertical spring stiffness k₃₃ and rotational stiffness k₅₅ considering two articulations for (a) reflection coefficient K _r , (b) transmission coefficient K _t , (c) shear force coefficients $S{F}^{*}$ , (d) bending moment coefficient $B{M}^{*}$ , (e) strain coefficient ${\tau }^{*}$ and (f) wave force coefficient K _FX .

Figure 24(c) to (e) demonstrate that the shear force, bending moment, and strain coefficients all rise with increasing vertical spring stiffness. A higher k₃₃ makes the joint stiffer in translation, effectively locking the two modules together vertically. This means that a wave force applied to module 1 is transmitted almost entirely through the stiff spring to module 2, rather than being absorbed by the relative motion of the joint. The efficient transfer of force results in greater shear forces at the joint and throughout the structure, allowing it to function more like a continuous, rigid body. In the case of high k₃₃, the modules are effectively immobilized in vertical motion. As the wave lifts one module but not its neighboring module, the stiff spring generates a significant reaction force that tries to keep the two modules level. This force, along with the geometry of the module, creates a substantial restoring moment at the joint, which increases the bending moment experienced by the structure. As a result, the entire structure bends as a single unit rather than allowing for independent motion that could relieve the moment. On the other hand, for lower values of k₃₃, the joint functions as a true decoupler, permitting considerable relative heave motion between the modules. This independence in movement means that wave forces are not efficiently transferred, which localizes the response. Consequently, this leads to lower global shear, bending moments, and strain on the structure, thereby enhancing its overall lifespan. However, this might also result in larger local deflections and more complex mooring requirements. Figure 24(f) illustrates that varying k₃₃ has little effect on the wave forces acting on the breakwater, as the primary contact occurs with the breakwater itself. As k₃₃ increases, the forces tend to be slightly higher, which can be explained by the increased rigidity of the system.

Conclusions

The hydroelastic behavior of VLFS integrated with porous floating breakwaters is analyzed considering multiple articulations in the VLFS, utilizing a coupled multi-domain boundary element method (MDBEM) and finite difference method (FDM). In order to ensure the reliability of the numerical framework, the developed numerical tool is validated against benchmark cases. Several critical parameters including breakwater porosity, mooring stiffness, and structural geometry are analyzed to determine their effects on shear forces ${\mathrm{SF}}^{*}$ , bending moments ${\mathrm{BM}}^{*}$ , strain in the floating elastic plate ${\tau }^{*}$ , and wave force coefficients K _FX . The key conclusions drawn from this study are as follows:

• The articulated joint acts as a primary energy dissipater. Articulations fragment the VLFS into smaller natural frequency bands, eliminating monolithic resonance peaks as seen from the results of surface deflections, bending moments, shear forces, and strain coefficient variations.

• The transmission coefficient K _t is noted to decrease uniformly with an increase in each articulation, highlighting the wave attenuation patterns. However, the reflection coefficients K _r tend to show decreasing patterns only for the case of VLFS with three articulations. Flexible VLFS deform in response to waves, dissipating energy through elastic hysteresis and enhancing reflection due to their structural compliance. On the other hand, stiff VLFS resist deformation, allowing more wave energy to pass beneath the structure, particularly for long-period swells, where flexibility plays a critical role. In summary, breakwater width primarily controls wave energy dissipation, while the stiffness of VLFS fine-tunes reflection.

• Lower material stiffness E reduces mid-hinge shear by 30%–40%, which is crucial for resilience against earthquakes and storms. Conversely, high modulus of elasticity E minimizes deflection but amplifies hydrodynamic vibrations.

• The combination of two or more parametric optimizations is preferred over a stand-alone optimization to ensure better wave energy attenuation characteristics of the structure to ensure ${\mathrm{SF}}^{*}$ and ${\mathrm{BM}}^{*}$ are localized within the inter-hinge regions.

• On providing articulation in VLFS shows overall reduction and localization of forces over the traditional monolithic VLFS constructions, ultimately helping in designing smaller and cheaper remedial measures, optimization of joint reinforcements and strain reduction measures.

• The study noted no significant changes in the wave forces K _FX with the increase in articulation, the patterns tend to lose the phase difference with the addition of each articulation. This shows that the waves become relatively calmer due to the increase in continuum length available for wave propagation.

The present 2D potential flow model does not account for three-dimensional edge effects, viscous dissipation in porous media and module gaps, or nonlinear wave-structure interactions. The mechanics of articulation are simplified, and environmental loading is restricted to regular wave patterns. Future research should focus on developing fully 3D coupled models that incorporate both viscous and inviscid flows (CFD-FEM/BEM) to better capture real flow physics and validate findings through experimental methods. Additionally, it is important to include nonlinear hinge models and conduct stochastic analyses under irregular wave spectra to predict connector fatigue and extreme responses. Ultimately, these improved models should serve as the foundation for a multi-objective optimization framework. This framework should simultaneously optimize articulation design, breakwater properties, and mooring configurations to create resilient and cost-effective VLFS for challenging offshore applications.