The spatial velocity potential function in the unbounded regions Ω1 and Ω2 can be expressed the eigenfunction expansion method [38] as follows (6)ϕΩ1=I0e−iq0(x+l0)ψ0(k0,z)+∑n=0∞Rneiqn(x+l0)ψn(kn,z) on ∂L, (7)ϕΩ2=∑n=0∞Tne−iqn(x−r0)ψn(kn,z) on ∂R,where ψn(kn,z) for n=0,1,2,… are defined by (8)ψn(kn,z)=(igω)cosh⁡kn(z+h)cosh⁡ (knh),and the dispersion relation is satisfied by kn, given (9)ω2=gkntanh⁡(knh).

The eigenfunctions ψn(kn,z) satisfy the orthogonal relation (10)⟨ψm(km,z),ψn(kn,z)⟩=Pnδmn,where δmn is the Kronecker delta function, and the normalization constant Pn is Pn=∫−h0ψn2(kn,z)dz=(−g2ω2)knh+sinh⁡ (knh)cosh⁡ (knh)2kncosh2⁡ (knh).

The undetermined coefficients Rn and Tn are calculated employing the orthogonality of the eigenfuction eigenfunction ψn in the outer domain as (11)Rn+δn0=1Pn∫−h0ϕΩ1(z)ψn(kn,z)dz, (12)Tn=1Pn∫−h0ϕΩ2(z)ψn(kn,z)dz.

The normal derivatives associated with the potential function are defined as (13)∂ϕΩ1∂n|x=−l0=−∂ϕΩ1∂x|x=−l0=iq0ψ0(k0,z)−∑n=0∞Rniqnψn(kn,z), (14)∂ϕΩ2∂n|x=r0=∂ϕΩ2∂x|x=r0=−∑n=0∞Tniqnψn(kn,z).

Substituting the unknown coefficients Rn and Tn in (13) and (14), respectively, after truncating the series to N terms, we obtain (15)∂ϕΩ1∂n|x=−l0=Q1[ϕΩ1]+2iq0ψ0(k0,z),where Q1[ϕΩ1] is defined by Q1[ϕΩ1]=−∑n=0N(iqnPn)(∫−h0(ϕΩ1(s)ψn(kn,s) ds))ψn(kn,z),

(16)∂ϕΩ2∂n|x=r0=Q2[ϕΩ2],where Q2[ϕΩ2] is given by Q2[ϕΩ2]=−∑n=0N(iqnPn)(∫−h0(ϕΩ2(s)ψn(kn,s) ds))ψn(kn,z).

These aforementioned two normal derivatives of the potential function constitute the boundary conditions for the left and right lateral boundaries.

Applying Green’s second identity to the spatial velocity potential ϕ(x,z) and the free-space Green’s function G(x,z;x0,z0) along the closed boundaries, we obtain (17)(−ϕ(x,z)12ϕ(x,z))=∫Γ(ϕ∂G∂n−G∂ϕ∂n)dΓ, (if (x,z)∈∂Γ if (x,z)∈Γ,(x,z)∉∂Γ),where the fundamental solution G that satisfies the Helmholtz equation (18)(∇2−ky2)G=δ(x−x0)δ(z−z0),where δ is the Dirac delta function. It is well known that the Laplace equation in cylindrical coordinates is given by, (19)∇2=∂2∂r2+1r∂∂r.

Substituting the above expression into (18), we obtain [39] ∂2G∂r2+1r∂G∂r−ky2G=δ,which gives r2∂2G∂r2+r∂G∂r−r2ky2G=δr2.

Now, let kyr=x so that r=xky. Substituting, we obtain x2ky2∂2G∂x2ky2+xkyky∂G∂x−x2ky2ky2G=x2ky2δ.

Finally, it reduces to x2∂2G∂x2+x∂G∂x−(x2−02)G=0.

The above equation corresponds to the general form of the Bessel equation. Its solution is given by G=J0(x) and G=K0(x).

The modified Bessel fuction of the first kind, J0(x) grows exponentially for large x, J0(x)≈ex asx→∞,while the modified Bessel function of the second kind, K0(x), exponentially for large x K0(x)≈e−x as x→∞.

Further, as the origin x→0, K0(x) has a logarithmic singularity given as K0(x)≈−ln(x)−γ+ln(2).

Thus, G=aK0(x), G=aK0(kyr) where a is a proportionality constant depending on the physical problem. Now, integrating (18) over a surface S, we obtain ∬S∇2G(r)ds−ky2∬SG(r)ds=∬Sδ(r)ds=1.

Using the divergence theorem, the first term can be expressed as ∬S∇2G(r)ds=∬S∇.(∇G)ds=∮∇G.n^dl.

Consequently, we further deduce that ∮∂G∂ndl=∮∂∂r(−ln(x)−γ+ln(2))ar^dl =−2πa.

Next, the second kind of equation is given by ky2∬SG(r)ds=ky2∫02π∫0ϵG(r)rdrdθ=2πky2∫0ϵG(r)rdr=2πaδ24[1−2γ+ln4−2ln(δ)].

The above expression tends to zero as x=0. Thus, we get −2πa−0=1 ⟹ a=−12π

Substituting the value of a into the solution gives G(x,z,x0,z0)=−12πK0(kyr),which is the fundamental solution of the Helmholtz equation, where r=(x−x0)2+(z−z0)2 is the Euclidean distance between the integration point (source point) (x0,z0) and the field point (x,z).

Further, the Bessel function of the recurrence formula is employed to determine the normal derivative of G, providing (20)∂G∂n=ky2πK1(kyr)∂r∂n,where K1 represents the modified Bessel function of the second kind and first order. The integration point (x0,z0) congruent with the field point (x,z), i.e., r→0 gets the asymptotic behavior (21)K0(kyr)=−γ−ln⁡(kyr2),where γ=0.5772 is the Euler constant. Substituting the boundary conditions (2)–(4) into the Green’s identity (17), then the integral equations in every boundary domain are obtained by (22)12ϕ(x,z)+∫ΓL(ϕ∂G∂n−G∂ϕ∂n)dΓ+∫ΓB(ϕ∂G∂n)dΓ+∫ΓR(ϕ∂G∂n−G∂ϕ∂n)dΓ+∫ΓF(∂G∂n−KG)ϕ   dΓ+∫ΓD(ϕ∂G∂n)dΓ=0.

To solve the aforementioned boundary integral equation, the region is discretized into a limited number of boundary elements, with assumed values ϕ and ∂ϕ constant across all elements (Wang and Meylan [40]). Furthermore, a set of linear algebraic systems is calculated using the point-collocation technique. The integral Eq. (22) is expressed as (23)∑j=1NLHijϕ−Gij∂ϕ∂n|ΓL+∑j=1NBHijϕ|ΓB+∑j=1NRHijϕ−Gij∂ϕ∂n|ΓR (24)+∑j=1NF(Hij−KGij)ϕ|ΓF+∑j=1NDHijϕ|ΓD=0.

Here, NL and NR are the number of elements along the lateral boundary, NB represents the number of elements along the bottom boundary, ND refers to the number of elements on the structure boundary, and NF indicates the number of elements on the free surface boundary. The terms Hij and Gijij are denote influence coefficients, which are defined as Hij=−12δij+∫Γj∂G∂ndΓandGij=∫ΓjGdΓ,when i≠j, the points (x,z) and (x0,z0) are located on various boundary elements. For this case, the required influence coefficients are calculated numerically using the Gauss-Legendre quadrature technique. On the other hand, when i=j, the field and integration points lie within the same boundary element, and the influence coefficients are evaluated analytically. The corresponding algebraic system is assembled into matrix form and evaluated using the Gauss elimination method to get the potential velocity and its normal derivatives on each boundary element. Furthermore, the reflection coefficient Kr, the transmission coefficient Kt, surface elevation η, vertical force FV, and horizontal force FH are calculated using the expressions provided in (25)Kr=|R0|, (26)Kt=|T0|.

The elevation of the free surface is represented by (27)η(x,0)=−iωg(ϕ(x,0).

The wave forces exerted on the breakwater are defined by (28)FH=−iωgh2∫ϕ(x,z)nxdz, (29)FV=−iωgh2∫ϕ(x,z)nzdx.

The convergence of the numerical solutions derived via BEM depends on the dimensions of the panels employed to discretize the boundaries in the closed domain. The panel size ps is determined as follows [40](31)ps=1κk0,where κ represents a constant of proportionality derived through convergence analysis. Table 1 illustrates the convergence behavior of the numerical solutions for different curved-leg geometries. From Table 1, it is evident that for κ=30 and ps=0.39, the computed wave reflection Kr and the transmission coefficient Kt converge accurately to the three decimal places.

The present section determines the wave reflection and transmission coefficients employing the boundary element method and compares them with previous research on rectangular structures. As the height of the curved-leg geometries approaches zero, the dual curved-leg pontoon floating structure effectively transforms into a rectangular configuration. Fig. 3a illustrates the reflection coefficient vs. the dimensionless height of the structure k0a. The present problem is reduced to the surface wave scattering by a rectangular structure, investigated analytically by Mei and Black [8] and experimentally in [41]. The present results are clearly shown to correlate with the analytical and experimental findings. The boundary element method is more accurate, as the results of the current problem agree well with those of previous studies.

Fig. 3b depicts the wave reflection Kr and the transmission coefficient Kt vs. for different models of the curved pontoon legs and the rectangle. The wave reflection increases by more than 5% for different dual curved-leg pontoon models when compared to the rectangular model, whereas the transmission decreases by more than 15%, as shown in Table 2. This attenuation of wave transmission is due to the attachment of the dual legs, which reduces transmitted wave energy by altering wave propagation patterns. Similar phenomena were observed in the T-shaped floating structure by Neelamani and Rajendran [10]. Also, Ruol et al. [14] discussed this for the π-shaped floating structure. Therefore, the wave-blocking performance of the curved legs is superior to that of the rectangular structure. A similar conclusion was drawn when comparing single-wing and box-type structures [23]. In addition, it is observed that the wave transmission of Model-2 increases significantly, whereas it decreases for Models 1, 3, and 4. This is due to the small area between the pontoon legs; as a result, higher wave transmission occurs. The transmission coefficient of Model-1 decreases, and the reflection increases relative to Models 2 and 3 due to the larger area between the legs, a phenomenon also observed in [24]. The flow direction changes due to the seaside of the concave leg. The same phenomenon was analyzed by Hussein et al. [21] while studying the concave and convex semi-circular barriers. Moreover, in Model 1, wave reflection increases by over 8% and wave transmission decreases by more than 27% compared to the rectangular model as shown in Table 2. This phenomenon is attributed to wave diffraction, which increases the wave height in front of the structure while decreasing it behind; a similar observation was analyzed in the experimental work on wing-shaped structures [42].

Fig. 4 presents the wave variation of wave-induced forces on the curved-leg pontoon breakwater vs. wave number k0h for various types of curved-leg models and a rectangular one. From Fig. 4a, it is noted that the horizontal force FH acting on the curved models increases compared to the rectangular structure. This higher force is attributed to wave slamming and breaking on the curved legs. This effect was reported by Ji et al. [43] when comparing wing-type and box-type structures. The force exerted on Model-1 increases because the seaside concave and leeside convex leg of the breakwater enhances wave trapping and reflection, resulting in greater reflected energy that increases the horizontal force. The vertical force FV acting on the rectangular structure increases relative to the other models, as shown in Fig. 4b. Additionally, the wave force FV for Model-1 and Model-4 eventually approaches zero due to the energy dissipation that occurs between the legs.

Fig. 5 illustrates the distribution of free surface elevation for different models and different heights of the curved-leg geometries. Fig. 5a depicts that the surface elevation of the rectangular model increases on both the seaside and leeside, whereas Model-1 shows a decrease on both sides. This reduction is attributed to the presence of the attached curved-leg pontoons, which increase the wave blocking effect. Furthermore, it is observed that the surface elevation on both the leeside and seaside of the breakwater is lower in Model-1 compared to other models due to the larger area between the legs. He et al. [24] reported a similar trend while comparing the different floating curved wing structures. From Table 2, it can be concluded that Model-1 is the most effective structure for attenuating waves compared to the other models. Therefore, Model-1 is considered more favorable for establishing a floating platform or supporting additional coastal activities. As a result, the rest of the discussion will focus primarily on Model-1. Fig. 5b shows that the surface elevation diminishes with increasing dr on the leeside of the structure due to the longer length of the legs, while it increases on the seaside.

Fig. 6 demonstrates the reflection coefficient Kr and the transmission coefficient Kt as a function of the wavenumber k0h for varying values of dr/h and da/h. Fig. 6a presents that wave reflection increases with increasing height of the curved-leg for k0h<1.5, and transmission is observed in the opposite trend for k0h<2. This is due to the blocking of waves by the larger curved-leg, which is similar to that noticed in [23] for a single-wing floating structure. Further, the wave transmission approaches zero. In Fig. 6b, for k0h<1.2, the wave reflection reduces as the height of the rectangular floating structure da/h decreases. It also illustrates that transmission decreases with increasing height da/h due to the more wave energy concentrated by the larger pontoon.

Fig. 7 demonstrates the wave reflection Kr and the wave transmission Kt vs. the wave number k0h for varying of b/h and l/h. In Fig. 7a, the reflection coefficient increases with increasing distance between the two legs because larger areas occur between the legs. In addition, the transmission decreases as b/h increases. This is due to the wave trapping between the legs. Fig. 7b shows that the wave reflection increases with increasing width of legs l/h and the transmission coefficient attains an opposite behavior. This can be attributed to the greater wave blocking by the larger width of the legs.

Fig. 8 illustrates the horizontal and vertical forces, FH and FV, excitation on the structure against the dimensionless wavenumber k0h for various heights of the legs dr/h. Fig. 8a shows the horizontal force FH increases to its maximum at k0h=0.8 and then gradually diminishes as the wavenumber k0h increases for all values of dr/h. As the leg height increases, the breakwater interacts with more wave energy, resulting in stronger wave loading and a corresponding rise in horizontal force. Fig. 8b reveals that as dr/h increases, the vertical force FV initially diminishes, reaching zero at k0h=1.6, and then gradually increases to attain stability due to enhanced wave pressure on the structure base.

Fig. 9 depicts the horizontal wave force FH and the vertical wave force FV as a function of the wavenumber k0h for varying rectangular height da/h and b/h, respectively. Fig. 9a demonstrates that the horizontal force FH attains a similar pattern as shown in Fig. 8a. It is noticeable that, for higher values of the da/h, the force increases with the larger area of the pontoon. Fig. 9b shows that for k0h<2, the vertical force FV increases as b/h increases due to the larger surface area between the legs that interacts with the incident wave, resulting in a higher wave force, and then zero force occurs at k0h=2.

Fig. 10 demonstrates the horizontal force FH and the vertical force FV as a function of the wavenumber k0h for the variation of l/h. In Fig. 10a, as the width of the dual curved-leg l/h increases, the horizontal force FH increases. This phenomenon arises from the effect of the slamming wave, resulting in an increase in the pressure on the structure. Further, Fig. 10b depicts that the vertical force FH reduces with increasing l/h, then reaches zero force at k0h=2.1 after FV increases. The reduction in force occurs because the larger width of the legs decreases the vertical pressure.

The variation of the wave reflection Kr and wave transmission Kt is plotted as a function of wave angle θ for different values of dr/h and b/h, as shown in Fig. 11. Fig. 11a depicts that the reflection coefficient increases with increasing incident angles θ and wave transmission exhibits the opposite behavior. This phenomenon can be attributed to the wave blocking by the larger legs. Further, in Fig. 11b, the transmission coefficient diminishes with an increase in distance between the two legs, and the reflection increases. This occurs because the wider rigid breakwater intercepts more of the incoming wave energy, thereby enhancing wave reflection and limiting the transmission beyond the structure.

Fig. 12 shows that the reflection coefficient Kr and transmission coefficient Kt against the wave angle θ for various values of da/h and l/h. Fig. 12a presents that Kr increases and transmission decreases, whereas for da/h increases owing to the larger height of the pontoon structure. On the other hand, in Fig. 12b, the reflection increases with increasing l/h. As l/h increases, the transmission decreases for θ<70∘ due to blockage of the waves.

Fig. 13 illustrates the horizontal force FH and vertical wave force FV against the wave angle θ for changes in dr/h. As shown in Fig. 13a, the horizontal force FH decreases as the dual leg height reduces because of the reduction of the wave reflection. Fig. 13b shows that the vertical force dramatically reduces to reach θ=90∘. Further, the vertical force decreases with increasing dr/h. This is because the greater wave energy is reflected and lesser energy is transmitted below the larger leg of the structure, leading to reduced vertical force.

Fig. 14 presents the wave forces on the structure FH and FV against θ for varying b/h. Fig. 14a shows that the horizontal force decreases with increasing b/h for θ<60∘ and increases for higher incident angles. The reduction of the wave force is due to the redirection of wave energy around the structure. Fig. 14b shows that the vertical force increases as the gap between the dual legs increases. As the larger surface area of the beneath structure interacts with the incident wave, it leads to a greater vertical force due to the higher pressure acting on the structure.

The surface plots showing the variation of horizontal force FH and vertical force FV, on the structure are presented in Fig. 15. In Fig. 15a, the horizontal force FH initially increases and reaches a maximum at k0h=1, after which it decreases monotonically as the wave number increases. The increase in wave force is due to higher wave reflection, as shown in Fig. 4. In addition, the FH becomes more stable with an increase in the incident wave angle θ and eventually approaches zero for larger angles. In Fig. 15b, as the k0h increases, the FV initially decreases; after attaining the zero force at k0h=2.1, it increases.