Wave-porous structure interaction is a critical area of study in coastal and ocean engineering, particularly in the design and analysis of coastal defenses such as breakwaters, submerged reefs, and other protective barriers. These porous structures typically composed of materials like rocks, concrete, or geotextiles and play a vital role in mitigating the impact of waves on shorelines, ports, and offshore installations. Understanding the wave interaction with porous media is essential for predicting the effectiveness in reducing wave energy, controlling erosion, and maintaining the stability of coastal environments. The interaction between waves and porous structures is a complex phenomenon that involves the transfer of energy, momentum, and mass between the fluid (wave) and the solid (porous structure). The presence of porosity within the structure introduces additional factors such as drag forces, wave attenuation, and dissipation, which must be accounted in the modeling and analysis of wave dynamics. These effects can significantly alter wave propagation, reduce wave height, and influence the wave-induced forces on the structure itself. Advanced computational models that combine fluid dynamics with porous media properties are essential for accurately capturing these interactions. Such models allow for a better understanding of the fluid flow through the porous material, the resistance it generates, and its impact on wave behavior. This knowledge is crucial for optimizing the design of coastal protection systems, ensuring their effectiveness, durability, and resilience to extreme wave events.

Numerical modeling of wave-porous structure interactions can be approached at either the microscopic or macroscopic scale. The microscopic approach involves modeling individual soil particles and analyzing the interaction of the wave with each particle. While this method can be accurate, it is computationally expensive and time-consuming. In contrast, the macroscopic model treats the porous structure as a whole, where wave energy is dissipated due to the collective drag forces of the soil particles. Although this approach may not provide precise force calculations on each individual particle, it allows for faster analysis of wave attenuation. This study focuses on wave attenuation, and therefore, the macroscopic model is used. The developed macroscopic model can be either mesh-based or meshless. In mesh-based approaches for wave–porous structure interactions, various numerical techniques have been adopted. The Finite Element Method was applied in [1], the Finite Difference Method in [2], and the Finite Analytic Method in [3]. A two-phase (fluid–air) model based on the Volume of Fluid (VOF) method was developed in [4]. The model proposed in [5] was enhanced by [6] using the VOF method to simulate wave overtopping of rubble-mound breakwaters. Applications of OpenFOAM in wave–structure interactions, including breakwater and scour modeling, are discussed in [7].

Several three-dimensional (3D) models have also been developed. For instance, Ref. [8] investigated dam-break wave interactions with porous structures using 3D volume-averaged Navier–Stokes (VANS) equations, incorporating turbulence through Large Eddy Simulation (LES). In [9], the VARANS equations were re-derived via a volume-averaging process, and the porous resistance coefficients were calibrated for dam-break cases. Subsequently, Ref. [10] developed the IHFOAM model, which solves the VARANS equations within the open-source framework OpenFOAM. All these models are based on a Eulerian framework.

In Lagrangian framework applications of the SPH method to porous media flows have been predominantly concentrated in civil engineering, particularly within hydraulic and coastal engineering disciplines. This emphasis is largely due to the practical challenges posed by fluid–structure interactions involving permeable coastal defenses and other porous barriers. Such problems are commonly associated with free-surface flows, a context in which SPH has proven to be especially effective. Several studies employing the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) approach have significantly advanced research in this field. For instance, the study in [11] investigated the width of the transition zone, while those in [12–16] validated WCSPH formulations for wave–current interactions involving porous media. Furthermore, the work in [17] extended the WCSPH method to simulate turbulent free-surface channel flows over porous gravel beds. In the context of turbulence modeling within SPH, the Sub-Particle Scale (SPS) approach is most commonly adopted, with occasional use of the k–ε model. The SPS method accounts for unresolved sub-grid-scale stresses at the particle level by employing an empirical model that relates these stresses to macroscopic flow variables [18–21]. In contrast, the k–ε model adopts a more complex framework involving transport equations for turbulent kinetic energy (k) and its dissipation rate (ε), but its performance is often sensitive to the calibration of several empirical constants [22–25]. Nevertheless, despite these developments, turbulence modeling within WCSPH still faces challenges, particularly in capturing flow behavior near porous boundaries, indicating the need for continued refinement and validation of existing models.

Parallel developments have also been made using incompressible SPH (ISPH) and moving particle semi-implicit (MPS) methods. These approaches, reported in studies such as [26–28] have been widely applied in coastal flow problems. Unlike WCSPH, ISPH and MPS solve Poisson’s equation to implicitly determine pressure, thereby yielding more accurate pressure fields—a key factor for resolving velocity and force distributions. However, these schemes are not without difficulties: configuring particle discretization so that the porous region has coarser spacing than the surrounding free-flow domain remains problematic. The abrupt change in resolution often induces spurious pressure oscillations near the interface, complicating numerical stability. Ref. [29] has reviewed all the available methods for wave porous structure interaction where the study concludes that the macroscopic model is well suited for coastal problems.

In our previous study [30], a two-dimensional wave–porous interaction model was developed using IMLPG_R, which demonstrated satisfactory performance. Similarly, Ref. [31] proposed a modified moving particle method that incorporated turbulence effects at the interface zone. Their findings showed that sharp velocity gradients at the interface can induce small turbulent jets, which were addressed through the implementation of a sub-particle scale (SPS) turbulence model to improve accuracy. In contrast, the present study extends our earlier work [30] by modeling the interface with a smooth velocity transition achieved through linearly varying porosity. This treatment reduces steep velocity gradients and suppresses the formation of small turbulent jets, and therefore turbulence effects are neglected in the current formulation. While these studies were primarily limited to two-dimensional cases, Ref. [32] also introduced a hybrid model to investigate wave–structure interactions. However, Lagrangian-based approaches remain relatively rare due to their high computational cost. More recently, Ref. [33] presented a three-dimensional Lattice Boltzmann model for simulating wave interaction with permeable structures with LES incorporated. This study shows the computational efficiency of the developed 3D Lattice Boltzmann method compared to other available SPH models.

In the present study, a hybrid model combining both mesh-based and mesh-free methods is employed to reduce computation time. This approach leverages the strengths of both models, enhancing accuracy. The FNPT model is irrotational and excludes viscous terms, while the 3D IMLPG_R model accounts for viscosity, incorporating additional drag forces from the soil particles. The larger domain is divided into two regions: the FNPT model is used in areas without the structure, and the 3D IMLPG_R model is applied to regions containing the porous structure. A buffer zone is introduced, where field variables from the FNPT model are transferred to the 3D IMLPG_R model. Fig. 1 gives an overview of the hybrid model. Ref. [32] demonstrated the efficiency of this hybrid model in simulating wave interactions with cylinders. This study extends their work by applying the model to wave-porous structure interactions. Ref. [32] shows the developed model for a flat-bottomed bed, which was later extended to include a sloping bottom in our previous study [34]. However, when the sloping bed was introduced, the particle distribution became uneven, and a significant gap appeared between the bottom bed and the fluid as the wave propagated. In our 2D IMLPG_R, the gaps between the particles are addressed using a minimum pressure gradient and interpolation as given in [35]. In 3D IMLPG, Ref. [32] adopted the re-distribution of the nodes. However, for complex geometry, redistribution of the nodes will not be effective. To address this issue, the Particle Shifting Algorithm (PST) was applied.

The study by [36] developed a PST algorithm for incompressible flows using Smooth Particle Hydrodynamics (SPH), building on the earlier work of [37]. In [38], various PST algorithms were reviewed, and their limitations were highlighted. Some of the reviewed PST algorithms include: the method proposed in [39], where the PST algorithm does not rely on Fick’s law but instead updates particle positions based on the velocity of neighboring particles, with consistency ensured by adding a correction term; and the approach in [37], where the PST algorithm is formulated on the basis of Fick’s law, and a velocity limit is imposed in deriving the concentration gradient. Ref. [40] shows a quasi-Lagrangian shifting where the influence of Mach number comes into picture. Ref. [41] developed a consistent approach of particle shifting which can be applied for compressible flow.

The generalized particle shifting technique proposed in [36] incorporates the dimensionality factor in the diffusion term, enabling seamless application to both two- and three-dimensional flows. Previous PST-based studies were often associated with high computational costs and challenges in selecting appropriate shifting coefficients, many of which were based on Mach number criteria and thus not suitable for the present study. Since the current model assumes incompressible flow, the PST algorithm of [36] was adopted and implemented with modifications to the weighting function and velocity ratio. This method was chosen because it is computationally efficient and provides a systematic approach for selecting shifting vectors, avoiding ad-hoc criteria. The PST algorithm works by redistributing particles from areas of high concentration to regions of lower concentration. This diffusion follows Fick’s law. The developed model along with the PST algorithm is used to validate for single porous layer flow and multi porous layer.

This paper is organized as follows; after the introduction section, the hybrid model is explained giving the governing equation for both 2D FNPT and IMLPG_R in Section 1. Section 2 gives the numerical algorithm followed in 3D IMLPG_R. Section 3 gives the problems caused due to the introduction of sloping bed and the PST algorithm applied. Section 4 gives the validation of developed model for single trapezoidal breakwater and two-layer trapezoidal breakwater. Section 5 gives the further studies done for solitary wave interaction with porous layer.

Governing equation for 2D FNPT:

FNPT assumes the fluid to be incompressible, irrotational and inviscid. The governing equation is the 2D Laplacian equation of velocity potential ϕ(x,z,t) and is given in Eq. (1). Here a 2D plane of XZ is used, where Z = 0 at the bottom and goes positive in the vertical direction. (1)∂2ϕ∂x2+∂2ϕ∂z2=0

The left boundary Г_w of the 2D domain is the wave maker. Neumann boundary condition given in Eq. (2a) is applied on the boundary Г_w. The bottom Г_b and the right boundary Г_r has the condition given in Eq. (2b). Free surface Г_f is assumed to be non-breaking and with non-linear dynamic free surface boundary condition derived from Bernoullis equation as given in Eq. (2c). (2a)∂ϕ∂x=xp˙(t)at x=xp˙(t) onΓw (2b)∂ϕ∂n=0onΓb andΓr (2c)∂ϕ∂t=−gη−12∇ϕ∇ϕonΓfwhere xp˙(t) is the wave probe velocity, g is the acceleration due to gravity and η is the free surface elevation. The initial condition is assumed as η=0 at time t = 0. Solution for this problem is obtained using Finite element method (FEM) with structured triangular mesh which has a variable spacing along the water depth. Ref. [42] explains the further details of this numerical model. An absorption layer is implemented upstream of the right boundary wall to attenuate the propagating wave and prevent reflections. In the present model, the length of the absorption layer is typically set to three times the wavelength of the incoming wave [43].

Governing equation for 3D IMLPG_R:

The governing formulation for three-dimensional porous flow within the IMLPG_R framework was originally established in the seminal work of [44], where the pressure gradient was demonstrated to vary linearly with the flow velocity through the introduction of the permeability coefficient—indicating that higher permeability facilitates greater fluid motion within the porous domain. Darcy’s law, which embodies this linear relationship, is applicable primarily to laminar flows characterized by low permeability. To extend its applicability to turbulent regimes, Ref. [45] incorporated a nonlinear drag term, wherein the resistance force varies with the square of the fluid velocity. Subsequently, Ref. [46] enhanced the model by including a virtual mass coefficient in the local acceleration term to account for added inertia effects.

In the context of wave–porous structure interactions, the resulting governing equations incorporate additional resistance components: a linear drag term representing laminar flow, a nonlinear drag term corresponding to turbulent flow, and a virtual mass term for inertial effects. The momentum equation thus attains a unified form, encompassing both linear and nonlinear drag contributions as expressed in Eq. (4). Following the approach proposed in [26], this formulation is implemented within a Lagrangian framework, wherein a single governing equation is applied seamlessly across both fluid and porous regions, with porosity serving as the distinguishing variable. The three-dimension model is based on Cartessian coordinates of OXYZ with Z = 0 and goes positive in the vertical direction. The velocity vector in three dimension is given as u→D=uDi^+vDj^+wDk^ (3)∇⋅u→D=0 (4)Crnw⋅Du→DDt=−1ρw∇→P+ϑeff∇2u→D−au→D−bu→D|u→D|here, C_r denotes the inertia coefficient, n_w represents the porosity, P is the pressure, and υ_eff corresponds to the effective viscosity as defined by Brinkman [47]. The coefficients a and b signify the linear and nonlinear drag terms, respectively, while g and ρ_w denote the gravitational acceleration and fluid density. This formulation provides a unified governing equation applicable to the entire computational domain, thereby eliminating the need for explicit interface treatment between the fluid and porous regions. (5)u→D=nwu→f (6)ϑeff=ϑnw

In this context, υ denotes the kinematic viscosity. The pressure at the free surface is considered to be zero (P = 0). The kinematic boundary condition, expressed in Lagrangian form, is given by dr→dt=u→f, where r→ represents the position vector and u→f denotes the velocity vector of the fluid particle.

On solid boundary, the following condition has to be satisfied, (7)n→⋅u→D=n→⋅U→D (8)n→⋅∇P=ρw⋅n→(ϑeff∇2u→D−a⋅u→D−b⋅u→D|u→D|+g→−U˙→D⋅Crnw)here, U˙→D denotes the acceleration of the solid boundary, and U→D represents its velocity. The linear and nonlinear drag coefficients, a and b, are functions of the Reynolds number and the prevailing flow regime. The linear drag component predominates under laminar conditions, whereas the nonlinear term becomes significant in turbulent flows. The generalized expressions for these coefficients are presented below, wherein the exponents k, m, and n vary depending on experimental observations reported in [48]. (9)a=αϑw(1−nw)knwmD50nb=β1−nwnw3D50Cr=1+1−nwnwγ with γ=0.34

The governing equation, together with the associated boundary conditions, is solved using the time-splitting algorithm proposed by Chorin [49], as detailed below,

(a) Compute the intermediate velocity by neglecting the pressure term in Eq. (4), yielding: (10)u→D∗=u→Dn+nw∗dtcr[ϑeff∇2u→D−au→D−bu→D|u→D|] (11a)x*=xn+uD∗dt (11b)y*=yn+vD∗dt (11c)z*=zn+wD∗dt

(b) The Pressure Poisson Equation (PPE) is solved using the intermediate velocity within a semi-implicit framework. (12)∇2Pn+1=cr∗ρwdt∗nw⋅∇u→D∗

(c) The pressure gradient at the n+1 time step is then employed to compute the Darcy velocity at the same step. (13)u→D∗∗=−dt∗nwρw∗cr∇Pn+1 (14)u→Dn+1=u→D∗−dt∗nwρw∗cr∇Pn+1+g→⋅dt

(d) Utilizing the Darcy velocity, the fluid particle velocities u_f, v_f and w_f in X, Y and Z directions, respectively, are determined from Eq. (4), after which the position vectors are updated accordingly based on the computed fluid velocities. (15)ufn+1=uDn+1nw;vfn+1=vDn+1nw;wfn+1=wDn+1nw (16)xn+1=xn+ufn+1dt;yn+1=yn+vfn+1dt;zn+1=zn+wfn+1dt

(e) The simulation is advanced iteratively until the desired end time is reached.

The Pressure Poisson Equation (Eq. (12)) is solved using the Improved Meshless Local Petrov–Galerkin (IMLPG_R) method. In this approach, the PPE is expressed in a weak formulation, with the pressure locally integrated over the domain. A Rankine source function is employed as the test function and applied to the PPE, as shown in Eq. (17). (17)φ=12πln(rRI)where RI is the radius of influence of the support domain. φ becomes zero at the boundary ∂Ω and satisfies ∇2φ=0 in domain ΩI excluding the center and the boundary ∂Ω. This is applied to the PPE and we get, (18)∫ΩI∇2PφdΩI=∫ΩIcr∗ρwnw∗dt∇u→D∗⋅φ⋅dΩI

Applying Gauss’s theorem, the Eq. (18) can be converted to surface integral as given below, (19)∫∂ΩInI→(P∇φ)dΩI−RIP=cr∗ρwnw∗dt∫ΩIu→D∗⋅∇φ⋅dΩI

The boundary conditions are applied in Eq. (19), where boundary, i.e., the velocity and pressure information from FNPT model, the solid boundary and the free surface boundary as given in Eqs. (7) and (8).

Now, the pressure P is discretized using the moving least square approximation as, (20)P(x→)≈∑J=1NϕJ(x→)⋅P^Jwhere, ϕ is the shape function. After applying the solid and free surface boundary conditions in Eq. (19a), the final matrix is given below, (21)[KIJ]⋅[P^]=[FI] (22a)KIJ={14π∫∂ΩIϕJ(x→)⋅dΩI−ϕJ(x→)forInnerfluidparticlen→⋅∇ϕJ(x→)forSolidBoundaryϕJ(x→)forfreesufaceparticles} (22b)FI={NC(I)dtRI4π∫ΩIu→D∗⋅dΩIforInnerfluidParticleNC(I)dtn→⋅(u→D∗−U→Dn+1)forsolidboundaryparticlePatmforfreesurfaceparticle}where, NC(I)=Crρw(I)nw(I).

In the stiffness matrix, Eq. (22a), the integral of the inner fluid particle is solved by a semi-analytical method, where a sphere is considered with numerical integration points assigned on the sphere. Eq. (23) is arrived based on [50] where the pressure is numerically integrated to the six points on the sphere. Moving Least Square (MLS) interpolation is used to get the values of the variables at the integration points. All the gradient forms are interpolated using SFDI [51]. (23)16(p1+p2+p3+p4+p5+p6)−p0=cr∗ρwnw∗dt∗RI12(uD1∗−uD3∗+vD2∗−vD4∗+wD5∗−wD6∗)

Fig. 2 gives the layout of the developed hybrid model, where the FNPT model is used to simulate the waves and get the input field variables for IMLPG_R. Meanwhile the IMLPG_R domain is kept ready with the background mesh consisting of porosity information. Once the input file from 2D FNPT is ready, it is applied at the buffer zone and further simulations are carried out. The 2D FNPT model field variables thus arrived will be in two dimensions. These field variables are further extended to third dimension (Y direction) in the IMLPG_R model, as we are interested in the uni-directional waves. The interpolation is based on [50].

Once the model is developed with PST algorithm, validation for wave porous structure interaction is carried out for which a single layer trapezoidal porous layer in a flat bottom bed has been considered. Then a second case of both porous sloping bed and impermeable sloping bed is simulated which is compared with experimental and numerical results.

This study presents a three-dimensional model developed to simulate wave interaction with porous structures. The approach weakly couples a two-dimensional Fully Nonlinear Potential Theory (FNPT) model with a three-dimensional Improved Moving Least Squares-based Point Galerkin (IMLPG-R) model where the turbulence effect is not considered. The developed model incorporates a smooth porosity transition across the interface, effectively reducing sharp velocity gradients in this region. This treatment suppresses the formation of localized turbulent jets and ensures consistency with the laminar flow assumption adopted in the present study. Previous SPH models for porous flow encountered two major challenges: (i) pressure or velocity fluctuations at the interface, and (ii) the need for iterative evaluation of porosity values at each time step. The present model addresses both issues. Within the IMLPG-R framework, the absence of a pressure gradient term in the stiffness matrix mitigates interface fluctuations, while the SFDI method provides porosity values at each time step directly, eliminating the need for iteration

The paper provides a detailed account of the mathematical formulation and the numerical algorithm employed. The primary focus is on the 3D IMLPG-R implementation, specifically its application to porous structures. While previous implementations of the 3D IMLPG-R model considered flat beds, this study extends it to include sloping beds. Incorporating the slope introduced several issues related to particle distribution in our 3D model, which were addressed using a Particle Shifting Technique (PST) based on the method proposed by [36]. The PST algorithm was optimized to best fit the developed model by fine-tuning several parameters such as the weight function, velocity ratio, constants, and radius of influence through a trial-and-error process. The importance of these parameters is investigated thoroughly in the present study.

The paper also describes the implementation of the numerical algorithm on a 3D background mesh and highlights challenges encountered when introducing an emerged trapezoidal porous structure. Three major challenges are discussed: maintaining uniformity in the force matrix, consistency in porosity and drag coefficient values, and accurate updating of the particle positions.

Furthermore, this study presents the modifications implemented in the three-dimensional (3D) numerical algorithm, extending the framework developed in the previous two-dimensional (2D) study. The model’s accuracy is verified through simulations of regular wave interactions with a single trapezoidal porous structure. Subsequent validations are performed for both porous and impermeable bed conditions, demonstrating good agreement with experimental results. It is noteworthy that the present investigation is limited to laminar flow, as the governing equations do not incorporate turbulence effects. The developed PST algorithm is further applied to a sloping bed configuration, where the earlier particle redistribution approach proved ineffective.