 Research
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 Published:
Investigation of offshore landslides impact on bucket foundations using a coupled SPH–FEM method
Geoenvironmental Disasters volume 11, Article number: 2 (2024)
Abstract
Background
As the exploitation of marine resources intensifies, the impact of submarine landslides on underwater structures has become a significant issue. Existing research primarily focuses on the impact on pipelines, often neglecting the actual deformation and mechanical response of underwater structures under impact loads in numerical simulations, thus complicating the evaluation of the reliability of these engineering structures in extreme conditions. Moreover, the dynamic response of bucket foundations, a common form of underwater base, under the effect of submarine landslide impacts remains unclear.
Methods
To address this knowledge gap, we have developed a fluidstructure coupling system that employs the coupled Smoothed Particle Hydrodynamics (SPH)Finite Element Method (FEM) to investigate a single impact process and analyze the displacement response of bucket foundations within a wateroffshore landslidebucket foundationsubgrade context. The accuracy of this developed method has been systematically verified through comparisons with previous experimental and numerical results.
Results
During a submarine landslide impact event, the impact force demonstrates a distinct decrease followed by stabilization, and the displacement response of the bucket foundation exhibits a rebound effect after reaching its maximum value. Furthermore, we conducted an extensive analysis of different impact angles for underwater data centers equipped with multibucket foundations. Our study revealed that groupbucket foundations experience a combined translationturnover failure when subjected to submarine landslide impacts, and the most unfavorable scenario for such impact is identified. The research introduces a novel numerical simulation approach for investigating the impact of submarine landslides on complex underwater structures.
Introduction
In recent years, the development of offshore wind power and oil platforms has led to significant advancements in bucket foundations. Compared to traditional offshore foundations, the bucket foundation possesses unique advantages during construction and operation (Gao et al. 2021), making them highly promising for various applications and attracting considerable attention from both academia and industry. However, more frequent extreme marine events are increasing geological hazards, such as seabed instability and submarine landslides caused by offshore storm surges, submarine earthquakes, hydrate decomposition, and human engineering activities (Sun and Bolin 2014; El Talibi et al. 2016; Nian et al. 2019; Khalfaoui et al. 2020; Shano et al. 2020; Zheng et al. 2021; Liu et al. 2023; Guo et al. 2023). These incidents have had a significant impact on underwater facilities in offshore engineering, leading to severe loss of life and property. For instance, in 1969, Hurricane Camille triggered a submarine landslide that damaged three offshore oil platforms in the Mississippi Delta, causing substantial economic losses (Bea 1971). In March 1977, an underwater oil pipeline belonging to Texaco in the United States was damaged by a submarine landslide, leading to a massive oil spillage (Camargo et al. 2019; Zhao et al. 2021). In 2006, 2009, and 2010, submarine landslides destroyed the underwater cables in the Luzon Strait multiple times, disrupting communications between Southeast Asian countries and China for up to 12 h (Hsu et al. 2008). Over the past 30 years, the Chinese Chengdao offshore oil field has experienced seven platform tilting incidents, seventeen cable failure incidents, and two submarine pipeline rupture incidents, all of which were speculated to be related to the destabilization of submarine sediments (Wang et al. 2020). Given the limited understanding of the mechanical behavior of bucket foundations under catastrophic loads of submarine landslides, it is crucial to conduct comprehensive research on the influence of offshore landslides on bucket foundations.
Currently, the study of landslide impact on underwater infrastructure heavily relies on numerical simulations, with Computational Fluid Dynamics (CFD) being widely used (Zakeri et al. 2009; Liu et al. 2015; Nian et al. 2018; Dutta and Hawlader 2019; Guo et al. 2019, 2021, 2022a, 2022b; Qian et al. 2020, 2023; Fan et al. 2022). In these simulations, various predictive models have been developed to assess the forces acting on underwater structures subjected to landslide impacts. However, these simulations often treat the structures (e.g., piles, pipelines) as fixed boundary walls, neglecting the actual deformation and mechanical response of underwater structures under impact loads, which makes it challenging to evaluate the reliability of these engineering structures under extreme conditions (Dong et al. 2017; Nian et al. 2018; Dutta and Hawlader 2019; Guo et al. 2019; Guo et al. 2022a, b; Qian et al. 2023). Additionally, these studies often simulate the instantaneous impact of homogeneous landslide bodies on underwater structures (Zakeri et al. 2009; Nian et al. 2018; Guo et al. 2021), neglecting the morphological evolution of the landslide's leading edge during the preimpact transportation process, which makes it deviates from the actual morphology of offshore landslides. In recent years, meshless methods, such as Smoothed Particle Hydrodynamics (SPH), have gained increasing popularity in geotechnical engineering studies (Dai et al. 2017; Zhang et al. 2019; Feng et al. 2022). The SPH method offers significant advantages in simulating the discontinuous deformation and largedeformation flow of soils (Chen and Yan 2021), as it does not rely on the assumption of macroscopic continuum media. On the other hand, the Finite Element Method (FEM) has achieved remarkable results in simulating structural deformations. Therefore, the coupling of these two methods within the Lagrangian framework can simulate the bidirectional fluid–structure coupling between underwater landslide (SPH particles) and bucket foundations (FEM). This allows for the determination of the dynamic displacement response of underwater foundations during landslide impact events. Furthermore, by incorporating Finite Element contact interactions between bucket foundations and the subgrade throughout the impact process, we can characterize the instability characteristics of underwater foundations subjected to impact loads. Hence, the SPHFEM coupling method exhibits substantial potential to emerge as a robust approach for addressing the aforementioned challenges. However, there is limited research specifically focusing on the application of the coupled SPHFEM method to investigate the impact of submarine landslides on underwater infrastructures (Luo et al. 2019).
To address this research gap, this study introduces the coupled SPHFEM method, establishes a threedimensional fluid–structure coupling system of the waterlandslide bodybucket foundationsubgrade, quantifies the dynamic response of the underwater foundation, and explores the effect of different variables on the bucket foundation and the underwater data center. The research aims to elucidate bucket foundations' operational performance and mechanical behavior when subjected to offshore landslides, ensuring safety and reliability throughout their service life.
Coupled SPHFEM method
Smoothed particle hydrodynamics method
As a fully Lagrangian method, the SPH method discretizes the original continuum by interpolating the values of variables, thus mitigating issues associated with mesh entanglement, distortion, and deformation in FE meshes. The SPH method utilizes integral interpolation to approximate the field variables f(x) (such as temperature, stress, density, etc.) at any given point. In this method, the value of a variable associated with a key particle x_{i} is approximated by taking the mean value of contributions from a set of neighboring particles x_{j} (Monaghan 1992), as given in Eq. (1).
where x_{i} is the focal particle, x_{j} is the neighboring particles, t is the time, V is the support domain; \(W(x_{i}  x_{j} ,h)\) is the kernel function, and h is the unit smooth length, defining the region of influence of the kernel function. The support domain is discretized into particles, and then a weighted summation is performed by considering all particles (Monaghan 1992), as given in Eq. (2).
where \(\Delta V_{j}\) is the volume occupied by particle j, which can be calculated from \({{m_{j} } \mathord{\left/ {\vphantom {{m_{j} } {\rho_{j} }}} \right. \kern0pt} {\rho_{j} }}\). Typically, it is necessary to differentiate the field variables (Monaghan 1992), as given in Eq. (3).
Since the variables f, m, and ρ are constant for a specific particle, solving for the gradient of the field variable can be transformed into solving for the gradient of the kernel function, thus avoiding the need for grid differencing. If the smoothing length for a particle is denoted as h, then the domain of the kernel function lies within a circle centered at the particle, with a radius of kh, which represents the smoothing length. The definition of the kernel function must satisfy various conditions. In this study, we illustrate the example of a cubic spline kernel (Monaghan 1994), as given in Eq. (4).
where \(x_{i}  x_{j}\) represents the relative distance between particles, defined by the particleization. The coupling between SPH particles is achieved by interpolation of the kernel function (i.e., SPH particles of different materials are considered as the same component, and a weighted sum is calculated for all particles within the smooth radius). To simulate the motion of fluids using the particle method, the motion state of particles needs to satisfy the fluid motion equations, namely the conservation of mass equation and the momentum equation (Monaghan 2002), which can be expressed in Eqs. (5), Eq. (6):
where v is the velocity, p is the pressure, g is the gravitational acceleration. Noting that \(\nabla_{i}\) adheres to the summation convention. The motion of fluid particles is induced by the pressure field gradient, governed by the equation of state. Taking an example of the linear UsUp Hugoniot form mentioned in this paper, the pressure p can be represented in Eq. (7):
where Us represents the shock velocity, Up is the particle velocity, η is the nominal volumetric compressive strain, c_{0} is the bulk speed of sound, as the intercept of the linear UsUp form and s is the slope, E_{m} is the initial value of specific energy and Γ_{0} is the Grüneisen ratio.
Coupling algorithm
In the SPHFEM coupling algorithm (shown in Fig. 1), a nodetosurface penalty contact algorithm is typically adopted to enforce balance and coordination conditions at the interface between the fluid and solid phases (Attaway et al. 1994; Campbell et al. 2000). Therein, SPH particles act as a nodebased surface with a contact thickness equal to the characteristic length l_{0}, while FE meshes serve as an elementbased surface. It is worth noting that l_{0} determines the collision volume size for each SPH particle. When discretizing the continuum, it is advisable to set l_{0} as half of the initial gap distance, ensuring that all particles are initially in exact contact, thus avoiding the pressure fluctuation effect caused by the initial particle overlay (Huang and Zhu 2015; Zhan et al. 2020). For each SPH particle in the fluid domain, the contact algorithm automatically checks for any penetration of a node into the main surface. Upon detecting penetration, a penalty contact force is introduced simultaneously to both SPH particles and FE meshes to ensure that fluid particles cannot penetrate the solid elements. The contact force F_{con} comprises both normal and tangential components, as given in Eqs. (8)–(10):
where \(\boldsymbol{F}_{\rm con}^{n}\) is the normal contact force, d is the penetration depth, k_{n} is the main surface stiffness factor, n is the normal vector of the element at the boundary, \(\boldsymbol{F}_{\rm con}^{\tau }\) is the tangential contact force, α is a coefficient, k is the friction coefficient, u_{FEM} is the displacement of nodes at the boundary, t is the time, and v is the velocity of SPH particles. The main surface stiffness factor k_{n} is related to the stable time Increment of the explicit computation Δt_{stable}, as given in Eq. (11).
where m_{i} is the mass of particles, and m_{n} is the nodal mass associated with the master segment. F_{con} establishes the correlation between the acceleration of SPH particles and the stress–strain of FE meshes. The stable time increment Δt_{stable} is given by:
where \(\Delta t_{\rm stable}^{\rm FEM}\) is the stable time increment of FE meshes, \(\Delta t_{\rm stable}^{\rm FEM}\) is the stable time increment of SPH particles, λ and λ' remain constant and are generally taken as 0.8 ~ 0.98, L is the length of elements, E is the elastic bulk modulus, υ is the passion ratio and ρ is the density. The velocity of SPH particles and the position of FE meshes in the subsequent increment are updated, to achieve coupling between the two methods. The coupling of the SPH method and FEM is executed within ABAQUS environment and its flowchart is shown in Fig. 2 (Liang and Chen 2019).
Validation of the numerical method
Validation of fluid motion patterns
To validate the proposed SPHFEM coupling method, a threedimensional fluid–structure interaction experiment [i.e., the ETSIN experiment (Lobovský et al. 2014)] is simulated. The experimental setup consisted of a water tank divided into two sections by a movable gate, as shown in Fig. 3. The tank's side walls and the gate were both made of PMMA plates. When the gate was lifted, a dambreak flow occurred, impacting the left side wall. This problem involved significant fluid deformation, splashing, and reflow after impact, which helped validate the accuracy and robustness of the numerical methods.
In the SPHFEM numerical model, the sidewalls are linearly elastic and modeled using FE meshes, with a density ρ = 1.19 g/cm^{3}, an elastic modulus E = 3.2 GPa, and a Poisson's ratio ν = 0.32 (Ucar et al. 2012). The fluid is modeled using 30256 SPH particles, with a characteristic length l_{0} = 5 mm and a cubic kernel function, which mitigates potential numerical instabilities (Xu and Kikuchi 2005). The smoothing length factor varies with the velocity field but is restricted to ensure that the number of particles associated with a single SPH particle is less than 140 for computational efficiency. The fluid follows the UsUp model, with a sound speed in water c_{0} = 1450 m/s and a dynamic viscosity μ = 8.87 × 10^{–4} Pa·s (Zhu and Fei 2022). Frictionless rigid contact is assumed between the fluid and the FE meshes.
The comparative results are presented in Fig. 4, showing that the experimental and numerical results are consistent in terms of the fluid's morphological characteristics. Additionally, the nondimensionalized position of the fluid leading edge z/H_{0} before impact is plotted in Fig. 5, comparing the proposed SPHFEM method with the original and modified Moving Particle Semiimplicit (MPS) method (Koshizuka and Oka 1996; Zhu et al. 2021). The results demonstrate that the SPHFEM method closely simulates the movement pattern of the fluid leading edge. Therefore, the utilization of the SPHFEM method for simulating free surface flow and highvelocity impact problems is feasible and yields accurate results. It's important to note that when simulating the 0.43s dam break process on a personal computer with an Intel i58300H CPU running at 3.8 GHz and 16 GB of RAM, it took 145 s of wallclock time (using 143098 increments) to complete the simulation with a stabilization time increment of t_{stable} = 2 × 10^{–6} s. The computation was achieved through a fourcore parallel computing framework within ABAQUS.
Validation of the landslide impact load
Furthermore, a quantitative validation of the fluid's impact load on the underwater pipeline (Dong et al. 2017) is conducted. In this case, the flowlike landslide has a density ρ = 1500 kg/m^{3}, with a length of 28 m and a height of 6 m. It moves with an initial velocity v_{0} = 6 m/s in the rightward direction, ultimately colliding with a fixed pipeline (shown in Fig. 6). The pipeline is smooth in contact, with a diameter D = 0.8 m and a burial depth of 0.4 m. The initial gap between the landslide body and the pipeline is 0.01 m.
In the coupled SPHFEM model, the rheological characteristics of the landslide are portrayed utilizing a HerschelBulkley model, wherein τ = 500 + 429.74 γ^{0.23} Pa. The characteristic length of the SPH particles is set as l_{0} = 0.05 m, making a total of 16,800 particles. The pressure p exerted on the pipeline is extracted and normalized to the dimensionless impact force p/(ρv^{2}), where v represents the current velocity. The resulting normalized impact force–time history curve is shown in Fig. 7, compared to the MPM method. The results demonstrate that the impact forcetime profile obtained from the SPHFEM simulation closely aligns with the MPM results, exhibiting errors in both peak and steadystate impact forces of less than 5%. Figure 8 displays the velocity vector distribution of particles near the pipeline at t = 0.12 s obtained by the proposed SPHFEM method, along with a comparison to the MPM method. The results demonstrate a strong agreement between the two methods in simulating the morphology of the landslide body. Based on the comparative analysis, it can be concluded that the coupled SPHFEM method proposed in this study is wellsuited for simulating the impact of underwater structures caused by submarine landslides.
Numerical analysis of landslide impact on bucket foundation
Modelling and simulated working conditions
Given the low occurrence rate of largescale submarine landslides in offshore environments, our study focuses on investigating smallscale landslides without subsequent sediment supply. On the basis of representing the morphology of the landslide's leading edge, the dynamic response of the bucket foundation is accurately examined. The landslide body, initially inclined at a slope angle of 45°, has a height of 1 m and a length of 8 m. It is positioned 2 m away from the foundation and exhibits an initial velocity of v_{0} = 5 m/s in the xdirection (as shown in Fig. 9b). The bucket foundation has a radius of 1 m, a burial depth of 2 m, a wall thickness of 20 mm, and a top plate positioned 0.5 m above the mud surface (as shown in Fig. 9c). The subgrade soil is assumed to be saturated and undrained cohesive soil due to its low permeability, and water pressure from the overlying water body is disregarded. The subgrade adopts an ideal elastoplastic constitutive model based on the Mohr–Coulomb failure criterion, with an undrained strength s_{u} = 9.54 kPa, an elastic modulus E_{s} = 500s_{u}, and a Poisson's ratio ν_{s} = 0.49. The remaining region of the model is filled with overlying water, which interacts with the landslide through interpolation using the kernel function. To accurately simulate the selfweight stress of the bucket foundation under normal service conditions and achieve a balance between internal and external water pressures, a surface load equal to the external water pressure is applied to the interior of the bucket foundation.
The steel bucket foundation is modeled as linear elastic with an average grid size of 0.02 m, while the subgrade adopts a grid size around the bucket of 0.01 m. The mechanical properties of the landslide body are quantified using the Herschel–Bulkley model (Coussot et al. 1998), while the ambient water is treated as a Newtonian fluid, following the UsUp model. Both the landslide body and water are modeled using SPH particles, with a characteristic length equal to 0.02 m. Cubic splines are employed as kernel functions, and the smoothing length factor is adjusted based on the expansion and compression trends of the model. To mitigate the occurrence of numerical oscillations in variables throughout the impact, the infinite element and Rayleigh damping are implemented. The numerical model parameters are summarized in Table 1 (Korson et al. 1969; Shi et al. 1999; Zakeri et al. 2008; Zhu and Fei 2022).
Results and analysis
Figure 10 depicts the time history curves of the impact force on the bucket foundation by the landslide body and the displacement response of the bucket foundation, wherein the impact force is calculated based on the contact force between the slide and the bucket foundation, and the displacement response is quantified by the maximum horizontal displacement (relative to the initial position) of the bucket foundation. Figure 11 shows the morphological characteristics and velocity vectors of the landslide body at different times.
It can be observed that during the motion of a landslide approaching a bucket foundation, the particles involved undergo collapse due to their selfweight, resulting in a decrease in the height and slope angle. When the leading edge of the landslide body makes contact with the bucket foundation (t = 0.35 s), the velocity of particles rapidly decreases, leading to an increase in impact force and displacement response. The particles flow over the top and around the sides of the bucket foundation. At t = 0.49 s, the force exerted on the foundation reaches its peak F_{peak} = 59 kN, followed by a maximum displacement response u_{stable} = 19 mm (at t = 0.51 s). As the kinetic energy of the landslide body dissipates, the velocity of the landslide decreases and stabilizes. At t = 1.00 s, the impact force on the bucket foundation gradually diminishes and eventually reaches a steady value of 5 kN, which can be considered as the soil pressure of the landslide mass deposited behind the foundation. The displacement response of the bucket foundation returns to 11 mm, due to the plastic deformation of the subgrade. At this stage, the impact behavior of the submarine landslide on the bucket foundation has essentially reached a stable state.
Discussions
Considering the uncertainty of offshore landslides, a variable parameter analysis is conducted by selecting 3 key indicators: velocity and thickness of landslide body and the shear strength of subgrade, aiming to uncover the impact effect patterns of offshore landslides on bucket foundations and to elucidate the underlying mechanisms influencing the disaster process.
Effect of velocity
By setting the initial velocities to 5 m/s, 10 m/s, and 20 m/s, the effect of velocity on the impact effect was studied. To ensure a consistent average thickness of the landslide at the moment of impact (H'), the initial spacing was adjusted accordingly. Figure 12 represents the evaluation of impact force, while Fig. 13 depicts the displacement response of the bucket foundation. As the velocity increases, both the impact force and displacement response show a substantial increase. For v_{0} = 5 m/s, the slides demonstrate blocking behavior, impeding the flow around the sides and top of the bucket foundation, resulting in a material accumulation at the rear of the bucket. During the process, the impact force exhibits a typical pattern of an increasepeakdecreasestabilization process. For v_{0} = 10 m/s, the flowlike slide around the bucket foundation maintains a higher velocity, resulting in a detachment of the landslide. The peak force duration is relatively long, and an additional secondary increase can be observed. For v_{0} = 20 m/s, the SPH particles disperse instantaneously, forming a continuous surface that envelops the bucket foundation. During the process, the maximum instantaneous impact force reaches 181 kN, with a brief duration and eventually oscillating near zero. Notably, the soil in the rear of the bucket experiences cracks development in the active zone, while the passive zone at the front bulges, resulting in the formation of a distinct spoonshaped penetration zone at the bottom, as shown in Fig. 13(a). Landslides with different velocities exhibit diverse motion behaviors, which significantly affect the pattern of impact force.
Effect of thickness
To investigate the effect of the relationship between the average thickness of the landslide and the height of the bucket foundation (i.e., less than, equal to, and greater than) on the impact process, landslides with different initial thicknesses H have been investigated. It should be noted that the initial slope angle and bucket foundation distance remain constant, and the average thickness H' of the landslide at the moment of impact is subsequently determined. Results are shown in Figs. 14 and 15. It is observed that when the average thickness H' of the landslide at the moment of impact is smaller than the height of the bucket, increasing the initial thickness of the landslide (from 0.5 to 1.0 m) leads to an increase in the maximum instantaneous impact force and displacement response of the foundation. Conversely, when the average thickness of the landslide H' at the moment of impact is greater than the height of the bucket, increasing the initial thickness of the landslide (from 1.0 to 1.5 m) no longer has a significant influence on the maximum instantaneous impact force or the maximum displacement response of the bucket foundation (both about only 3% increase). However, the increased total volume and kinetic energy of the landslide prolong the impact force duration, resulting in a larger stable displacement of the bucket foundation (20% increase). Given that the stability of bucket foundations in extreme conditions is primarily assessed based on their displacement response, future research should pay more attention to this aspect.
Effect of strength
Considering the uncertainty regarding the location of underwater landslides, a parametric analysis has been conducted for the undrained shear strength s_{u} of the subgrade soil, where s_{u} varies from 5 to 20 kPa. The initial velocity of the landslide body is set to 10 m/s (initial spacing is adjusted accordingly), and the model's other parameters are consistent with those described in 4.1. Figures 16 and 17 respectively illustrate the timehistory curves for impact forces and bucket foundation displacement responses. It is evident that the increase in the undrained shear strength of the subgrade soil has no significant impact on the direct impact forces experienced by the bucket foundation. However, it notably influences the horizontal displacement of the bucket foundation when the impact force exceeds the subgrade's impactbearing capacity limit. With the increase in subgrade soil strength (s_{u} ranging from 5 to 20 kPa), the maximum horizontal displacement of the bucket foundation changes from 24 mm to 0.2 m. Correspondingly, the foundation experiences instability, leading to overturning and failure under the impact. These findings further emphasize the advantages of the proposed method (SPHFEM coupling method) in simulating fluid–structure coupling impact problems. It can model the interaction between the bucket and soil throughout the bidirectional fluid–structure impact, as well as quantify the dynamic displacement response of the bucket foundation during the impact process.
A case study: offshore landslides impact on the underwater data center
In offshore engineering, group bucket foundations are commonly employed for various purposes such as offshore wind turbines and underwater data centers. According to relevant reports (Simon 2018; Sutherland and Bopp 2023), Microsoft has successfully deployed an underwater data center using a threebucket configuration (shown in Fig. 18a). The innovation offers advantages such as reduced energy consumption, lower carbon emissions, improved computer performance, and support for environmentally friendly and sustainable development of data centers. With the increasing demand for largescale scientific computing, the establishment of underwater data centers has become a crucial strategic need requirement. Therefore, conducting numerical simulations on the impact of offshore landslideinduced on underwater data center foundations is crucial for optimizing the design of such infrastructures.
Modelling and simulated working conditions
The numerical underwater data center model consists of a steel pressure vessel, a steel frame (steel support plates, steel beams), and three steel bucket foundations. The upper part of the facility is the pressure vessel, with a total length of 12.2 m, an inner diameter of 2.8 m, an outer diameter of 3.18 m, and an average density of 2500 kg/m^{3}. The middle part of the facility is the steel beams' structure, measuring 14.3 m in length and 12.7 m in width. The steel beams each have a height of 500 mm, a width of 450 mm, a flange thickness of 25 mm, and a web plate thickness of 25 mm. The lower part of the facility consists of three connected bucket foundations. Each bucket is numbered, with a radius of 1.0 m, a burial depth of 2.0 m, and a total height of 2.5 m (shown in Fig. 18b). The subgrade adopts an elastoplastic constitutive model, as outlined in 4.1, and a grid size around the bucket of 0.01 m. All pertinent physical and mechanical parameters of the overlying water body, as well as the contact details within the model, remain consistent with those described in 4.1.
To account for the uncertainty of the landslide's impact direction, three directions with relatively large projected lengths (L_{p}) labeled A, B, and C are proposed, to investigate the effect of impact direction on the displacement response of the data center. The offshore landslide is assumed to have an initial thickness of 2 m, a slope angle of 45°, and an extension length of 50 m (three times more than L_{p}). The initial density of the landslide is ρ = 1681 kg/m^{3}, and the initial velocity is v_{0} = 10 m/s. The characteristic length of the landslide's SPH particles is 0.05 m, while the characteristic length of the water particles is 0.1 m. Figure 19 depicts the results of the landslide impact on the underwater data center.
Results analysis and suggestions
Figure 19b reveals that in Direction A and B, both the maximum instantaneous impact force and the corresponding foundation displacement response are significantly larger compared to Direction C. The influence of directions on the maximum displacement response can reach up to 80%. The substantial volume of the landslide causes the maximum instantaneous impact force to exceed the bearing capacity of the foundation, resulting in instability failure of the group bucket foundation in all cases. Additionally, the large size of the structure leads to inadequate flow around the landslide, prolonging the duration of the impact force compared to a single bucket and resulting in a greater displacement response. The group bucket foundation experiences a combined effect of horizontal forces and overturning moments from the landslide of Direction A, ultimately leading to combined translationturnover failure, characterized by an uplift of Bucket 1 and 3 and a downward of Bucket 2, as shown in Fig. 20.
In the context of designing underwater data centers, it is essential to prioritize the assessment of the foundation's resistance under Direction A (i.e., when Bucket 1 and 3 or Bucket 2 and 3 are simultaneously subjected to forces). Special attention should be given to the deformation of the steel beams and the reliability of the steel frame to prevent connection failures caused by offshore landslides. Moreover, the efficient removal of deposited sediment from the impact flow is crucial for ensuring effective heat dissipation. It is worth noting that, due to the absence of detailed model specifics and onsite measurement data for underwater data centers, the aforementioned numerical simulations were conducted under several assumptions. Nonetheless, the proposed methods still aid in recognizing the most critical scenarios and the pattern of foundation instability and the qualitative conclusions derived can also serve as recommendations for the design of underwater data centers.
Conclusions
The impact of offshore landslides on underwater foundations was investigated using the proposed SPHFEM method, focusing on analyzing the impact process and dynamic response of bucket foundations. The effects of the initial velocity, thickness, shear strength of subgrade and direction of the landslide on the impact of the bucket foundation and underwater data center were examined, leading to the following conclusions.

1.
The proposed SPHFEM method is well suited for simulating the submarine landslide impact on bucket foundations. In a single impact process, the impact force demonstrates a distinct decrease followed by stabilization, while the displacement response of the bucket foundation shows a rebound effect after reaching its maximum value.

2.
Lowspeed slides (v_{0} = 5 m/s) exhibit blocking and accumulation behaviors, whereas highspeed slides (v_{0} = 20 m/s) display a typical disperse behavior. The increase in thickness and shear strength of subgrade soil does not always affect the magnitude of the peak impact force directly, instead, it contributes to a larger stable displacement response.

3.
The groupbucket foundations of the underwater data center undergo combined translationturnover failure under the landslide impact. The impact direction of landslides also plays a significant role in the impact process. Specifically, Direction A represents the most unfavorable scenario for offshore landslide impact on the underwater data center.
Availability of data and materials
All data, models, and code generated or used during the study appear in the submitted article.
Abbreviations
 CFD:

Computational fluid dynamics
 FE:

Finite element
 FEM:

Finite element method
 MPM:

Material point method
 PMMA:

Polymethyl methacrylate
 SPH:

Smoothed particle hydrodynamics
References
Attaway SW, Heinstein MW, Swegle JW (1994) Coupling of smooth particle hydrodynamics with the finite element method. Nucl Eng Des 150(2–3):199–205. https://doi.org/10.1016/00295493(94)901368
Bea RG (1971) How sea floor slides affect offshore structures. Oil Gas J 69(48):88–92
Camargo JM, Silva MV, Ferreira Júnior AV, Araújo TC (2019) Marine geohazards: a bibliometricbased review. Geosciences 9(2):100. https://doi.org/10.3390/geosciences9020100
Campbell J, Vignjevic R, Libersky L (2000) A contact algorithm for smoothed particle hydrodynamics. Comput Methods Appl Mech Eng 184(1):49–65. https://doi.org/10.1016/S00457825(99)004429
Chen F, Yan H (2021) Constitutive model for solidlike, liquidlike, and gaslike phases of granular media and their numerical implementation. Powder Technol 390:369–386. https://doi.org/10.1016/j.powtec.2021.05.023
Coussot P, Laigle D, Arattano M, Deganutti A, Marchi L (1998) Direct determination of rheological characteristics of debris flow. J Hydraul Eng 124(8):865–868. https://doi.org/10.1061/(ASCE)07339429(1998)124:8(865)
Dai Z, Huang Y, Cheng H, Xu Q (2017) SPH model for fluid–structure interaction and its application to debris flow impact estimation. Landslides 14(3):917–928. https://doi.org/10.1007/s1034601607774
Dong Y, Wang D, Randolph MF (2017) Investigation of impact forces on pipeline by submarine landslide using material point method. Ocean Eng 146:21–28. https://doi.org/10.1016/j.oceaneng.2017.09.008
Dutta S, Hawlader B (2019) Pipeline–soil–water interaction modelling for submarine landslide impact on suspended offshore pipelines. Géotechnique 69(1):29–41. https://doi.org/10.1680/jgeot.17.P.084
El Talibi H, El Moussaoui S, Zaghloul MN, Aboumaria K, Wassmer P, Mercier JL (2016) New sedimentary and geomorphic evidence of tsunami flooding related to an older events along the TangierAsilah coastal plain, Morocco. Geoenviron Disasters 3:1–21. https://doi.org/10.1186/s4067701600496
Fan N, Zhang W, Sahdi F, Nian T (2022) Evaluation of horizontal submarine slide impact force on pipeline via a modified hybrid geotechnical–fluid dynamics framework. Can Geotech J 59(6):827–836. https://doi.org/10.1139/cgj20210089
Feng R, Fourtakas G, Rogers BD, Lombardi D (2022) Twophase fullycoupled smoothed particle hydrodynamics (SPH) model for unsaturated soils and its application to rainfallinduced slope collapse. Comput Geotech 151:104964. https://doi.org/10.1016/j.compgeo.2022.104964
Gao B, Ye G, Zhang Q, Xie Y, Yan B (2021) Numerical simulation of suction bucket foundation response located in liquefiable sand under earthquakes. Ocean Eng 235:109394. https://doi.org/10.1016/j.oceaneng.2021.109394
Guo X, Fan N, Liu Y, Liu X, Wang Z, Xie X, Jia Y (2023) Deep seabed mining: frontiers in engineering geology and environment. Int J Coal Sci Technol 10:23. https://doi.org/10.1007/s4078902300580x
Guo X, Nian T, Fan N, Jia Y (2021) Optimization design of a honeycombhole submarine pipeline under a hydrodynamic landslide impact. Mar Georesour Geotechnol 39(9):1055–1070. https://doi.org/10.1080/1064119X.2020.1801919
Guo X, Nian T, Stoesser T (2022a) Using dimples to reduce submarine landslide impact forces on pipelines at different span heights. Ocean Eng 244:110343. https://doi.org/10.1016/j.oceaneng.2021.110343
Guo X, Stoesser T, Nian T, Jia Y, Liu X (2022b) Effect of pipeline surface roughness on peak impact forces caused by hydrodynamic submarine mudflow. Ocean Eng 243:110184. https://doi.org/10.1016/j.oceaneng.2021.110184
Guo X, Zheng D, Nian T, Yin P (2019) Effect of different span heights on the pipeline impact forces induced by deepsea landslides. Appl Ocean Res 87:38–46. https://doi.org/10.1016/j.apor.2019.03.009
Hsu SK, Kuo J, ChungLiang L, ChingHui T, Doo WB, Ku CY, Sibuet JC (2008) Turbidity currents, submarine landslides and the 2006 Pingtung earthquake off SW Taiwan. Terr Atmos Ocean Sci 19(6):7. https://doi.org/10.3319/TAO.2008.19.6.767(PT)
Huang Y, Zhu C (2015) Numerical analysis of tsunami–structure interaction using a modified MPS method. Nat Hazards 75:2847–2862. https://doi.org/10.1007/s1106901414641
Khalfaoui O, Dezileau L, Degeai JP, Snoussi M (2020) A late Holocene record of marine highenergy events along the Atlantic coast of Morocco: new evidences from the Tahaddart estuary. Geoenviron Disasters 7(1):1–14. https://doi.org/10.1186/s40677020001695
Korson L, DrostHansen W, Millero FJ (1969) Viscosity of water at various temperatures. J Phys Chem 73(1):34–39. https://doi.org/10.1021/j100721a006
Koshizuka S, Oka Y (1996) Movingparticle semiimplicit method for fragmentation of incompressible fluid. Nucl Sci Eng 123(3):421–434. https://doi.org/10.13182/NSE96A24205
Lobovský L, BotiaVera E, Castellana F, MasSoler J, SoutoIglesias A (2014) Experimental investigation of dynamic pressure loads during dam break. J Fluids Struct 48:407–434. https://doi.org/10.1016/j.jfluidstructs.2014.03.009
Liang S, Chen Z (2019) SPHFEM coupled simulation of SSI for conducting seismic analysis on a rectangular underground structure. Bull Earthq Eng 17:159–180. https://doi.org/10.1007/s105180180456z
Liu J, Tian J, Yi P (2015) Impact forces of submarine landslides on offshore pipelines. Ocean Eng 95:116–127. https://doi.org/10.1016/j.oceaneng.2014.12.003
Liu X, Wang Y, Zhang H, Guo X (2023) Susceptibility of typical marine geological disasters: an overview. Geoenviron Disasters 10(1):1–31. https://doi.org/10.1186/s40677023002376
Luo G, Pan S, Zhang Y, Chen L (2019) Response analysis of submerged floating tunnel hit by submarine based on smoothedparticle hydrodynamics. Shock Vib 2019:1–12. https://doi.org/10.1155/2019/9056416
Monaghan JJ (1992) Smoothed particle hydrodynamics. Ann Rev Astron Astrophys 30(1):543–574. https://doi.org/10.1146/annurev.aa.30.090192.002551
Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399–406. https://doi.org/10.1006/jcph.1994.1034
Monaghan JJ (2002) SPH compressible turbulence. Mon Not R Astron Soc 335(3):843–852. https://doi.org/10.1046/j.13658711.2002.05678.x
Naval Group (2018) Project Natick: Naval Group collaborates with Microsoft to deploy underwater Datacenter. France. https://www.youtube.com/watch?v=hSIaC_TgvyI
Nian T, Guo X, Fan N, Jiao H, Li D (2018) Impact forces of submarine landslides on suspended pipelines considering the lowtemperature environment. Appl Ocean Res 81:116–125. https://doi.org/10.1016/j.apor.2018.09.016
Nian T, Guo X, Zheng D, Xiu Z, Jiang Z (2019) Susceptibility assessment of regional submarine landslides triggered by seismic actions. Appl Ocean Res 93:101964. https://doi.org/10.1016/j.apor.2019.101964
Qian X, Liu Z, Xu J (2023) New insights into quantifying the stable impact forces of a submarine debris flow on a pipeline. Ocean Eng 274:114060. https://doi.org/10.1016/j.oceaneng.2023.114060
Qian X, Xu J, Bai Y, Das HS (2020) Formation and estimation of peak impact force on suspended pipelines due to submarine debris flow. Ocean Eng 195:106695. https://doi.org/10.1016/j.oceaneng.2019.106695
Shano L, Raghuvanshi TK, Meten M (2020) Landslide susceptibility evaluation and hazard zonation techniques—a review. Geoenviron Disasters 7(1):1–19. https://doi.org/10.1186/s40677020001520
Shi X, Xu R, Gong X, Chen G, Yuan Z (1999) Experimental study on horizontal bearing capacity of single bucket foundation. Chin J Geotech Eng 06:723–726 (in Chinese)
Simon K (2018) Project NatickMicrosoft’s selfsufficient underwater datacenters. IndraStra Global 4(6):4
Sun Y, Bolin H (2014) A potential tsunami impact assessment of submarine landslide at Baiyun Depression in Northern South China Sea. Geoenviron Disasters 1(1):1–11. https://doi.org/10.1186/s4067701400070
Sutherland T, Bopp G (2023) The Pacific futures of subsea data centers. New Media Soc 25(2):345–360. https://doi.org/10.1177/1461444822114994
Ucar Y, Akova T, Aysan I (2012) Mechanical properties of polyamide versus different PMMA denture base materials. J Prosthodont Implant Esthetic Reconstr Dent 21(3):173–176. https://doi.org/10.1111/j.1532849X.2011.00804.x
Wang Z, Sun Y, Jia Y, Shan Z, Shan H, Zhang S, Wen M, Liu X, Song Y, Zhao D, Wen S (2020) Waveinduced seafloor instabilities in the subaqueous Yellow River Delta—initiation and process of sediment failure. Landslides 17:1849–1862. https://doi.org/10.1007/s10346020013992
Xu F, Kikuchi M (2005) The study of kernel functions in impact problem simulation by using SPH method. Int J Comput Methods 2(04):587–600. https://doi.org/10.1142/S0219876205000612
Zakeri A, Høeg K, Nadim F (2008) Submarine debris flow impact on pipelines—Part I: experimental investigation. Coast Eng 55(12):1209–1218. https://doi.org/10.1016/j.coastaleng.2008.06.003
Zakeri A, Høeg K, Nadim F (2009) Submarine debris flow impact on pipelines—part II: numerical analysis. Coast Eng 56(1):1–10. https://doi.org/10.1016/j.coastaleng.2008.06.005
Zhan L, Peng C, Zhang B, Wu W (2020) A SPH framework for dynamic interaction between soil and rigid body system with hybrid contact method. Int J Numer Anal Methods Geomech 44(10):1446–1471. https://doi.org/10.1002/nag.3070
Zhang W, Zheng H, Jiang F, Wang Z, Gao Y (2019) Stability analysis of soil slope based on a water–soilcoupled and parallelized Smoothed Particle Hydrodynamics model. Comput Geotech 108:212–225. https://doi.org/10.1016/j.compgeo.2018.12.025
Zhao E, Dong Y, Tang Y, Cui L (2021) Numerical study on hydrodynamic load and vibration of pipeline exerted by submarine debris flow. Ocean Eng 239:109754. https://doi.org/10.1016/j.oceaneng.2021.109754
Zheng D, Lei D, Yan C, Shen Y, Nian T (2021) Global research trends in submarine landslides: a bibliometric analysis based on Web of Science publications. J Eng Geol 29(6):1805–1814. https://doi.org/10.13544/j.cnki.jeg.20210712. (in Chinese)
Zhu C, Fei Q (2022) Impact characteristics and strength evaluation of a slender body obliquely entering water at high speed. J Phys Conf Ser 2338(1):012–024. https://doi.org/10.1088/17426596/2338/1/012024
Zhu C, Chen Z, Huang Y (2021) Coupled moving particle simulation–finiteelement method analysis of fluid–structure interaction in geodisasters. Int J Geomech 21(6):04021081. https://doi.org/10.1061/(ASCE)GM.19435622.0002041
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The authors thank all the scholars in the references for their contributions in their research fields.
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Funding for the research has been supported by the National Natural Science Foundation of China (42077272, 42377185 and 52079020), the LiaoNing Revitalization Talents Program (XLYC2002036), the open research fund program of Zhoushan Field Scientific Observation and Research Station for Marine Geohazards, China Geological Survey (No. ZSORS22–04/09), and the Opening fund of State Key Laboratory of Coastal and Offshore Engineering at Dalian University of Technology (LP2310).
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ZW contributed to the implementation, software, and writing—original draft preparation. DZ contributed to the conceptualization and writing—review & editing. XG contributed to the modeling and writing—review & editing. ZG contributed to the writing—review & editing. YS contributed to the writing—review & editing. TN contributed to the conceptualization, design and writing—review & editing. The authors read and approved the final manuscript.
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Wang, Z., Zheng, D., Guo, X. et al. Investigation of offshore landslides impact on bucket foundations using a coupled SPH–FEM method. Geoenviron Disasters 11, 2 (2024). https://doi.org/10.1186/s40677024002669
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DOI: https://doi.org/10.1186/s40677024002669