Finite Element Discretization Research Articles

Advanced tensile fracture analysis of alumina ceramics: Integrating hybrid finite-discrete element modeling with experimental insights

This study offers novel insights into the Mode I tensile response of an alumina ceramic through the use of computational modeling and the flattened Brazilian disk (FBD) experiments. A modified hybrid finite-discrete element method (HFDEM) is developed, integrating a coupled damage and friction cohesive model and a microscopic stochastic fracture model with a Weibull strength distribution by Monte Carlo simulation. The model is used to simulate direct tensile failure processes under quasi-static loading conditions, providing qualitative and quantitative predictions of direct tensile failure processes of an alumina ceramic. Concurrently, quasi-static flattened Brazilian disk tests (indirect tensile tests) are performed on a standard MTS machine coupled with a high-speed camera. The modified HFDEM model is also applied to reproduce the FBD experiments, and our simulated tensile strength is consistent with the experimental results. The results of the modified HFDEM model show three kinds of phenomena (i.e., “underestimation”, “reasonable estimation”, and “overestimation” of the indirect tensile strength) and four different types of associated fracture and fragment patterns of FBD testing. The integration of simulation and experimental results reveal relationships between fracture patterns, fragment geometry, tensile strength, and indirect tensile strength. The fracture and fragmentation patterns derived from our modified HFDEM model can be utilized to analyze the “tensile strength” measured in BD testing. Overall, this research offers important insights and direction for future Brazilian disk experiments and tensile strength assessments, enhancing our comprehension of ceramic failure mechanisms under tensile loadings.

Finite Element Discretizations of a Convective Brinkman–Forchheimer Model Under Singular Forcing

Finite Element Discretizations of a Convective Brinkman–Forchheimer Model Under Singular Forcing

Effects of Geo-Stress and Natural Fracture Strength on Hydraulic Fracture Propagation in a Deep Fractured Tight Sandstone Reservoir.

Hydraulic fracturing technology has been widely used in the tight reservoir reconstruction. Unfortunately, with the deepening of mining depth and the increase of geo-stress, the propagation mechanism of medium-pressure fractures in the reservoir is significantly different from that of conventional shallow reservoirs. Based on the combined finite discrete element method, this paper conducts numerical simulation research on deep tight sandstone reservoirs in the west. The discrete fracture network modeling method is used to establish a tight sandstone reservoir model with natural bedding, and the influence of geo-stress difference and natural fracture strength on hydraulic fracture propagation law in a high geo-stress environment is discussed in detail. The results show that the difference between geo-stress and the strength of natural fractures has a significant effect on the shape and expansion of hydraulic fractures under the high geo-stress conditions. The greater the difference in ground stress, the more obvious the tendency of the main fractures of the reservoir, and the shorter the branch fractures. With the increase of natural fracture strength, the changes in propagation pressure, fracture length, area, and width, which can be fitted with a linear function with a goodness of fit as high as 0.99. In addition, the morphological results of hydraulic fractures in the simulation are not only affected by the constitutive parameters of the model but also may be affected by the randomness of the natural fracture network, thus, showing a certain degree of dispersion. Therefore, it is extremely necessary to build a reservoir fracturing model in a specific area based on more detailed geological monitoring data to guide actual construction. The above achievements have certain reference significance for the field operation of deep tight sandstone reservoirs.

Enhanced heat transfer in vented square enclosures with block structures: Exploring iso-perimetric shapes and multigrid approach for mixed convection of nano-fluids

A Multigrid approach was employed to examine the fluid flow and heat transfer features of a nano-fluid undergoing mixed convection within a vented enclosure featuring a variety of iso-perimetric-shaped obstacles. Concerning the cavity, the inlet port is situated at the lower left corner, while the outlet port is located at the upper right corner. The study involved conducting simulations with a range of control parameters by employing the higher-order mixed finite element technique. To capture key physical processes like mass transfer, fluid flow, and heat transfer within the model, we employed a numerical technique called finite element discretization. Specifically, we used a stable combination of cubic and quadratic (P3 ∕P2) approximations on a hybrid grid to discretize the governing equations, ensuring accurate and reliable results. The study examines the influence of governing parameters within specified ranges, including nanoparticle volume fraction (0 % ≤ φ ≤ 10 %), the Richardson number (0.01 ≤ Ri ≤ 5), and obstacle positions (0.3 ≤ C ≤ 0.7). Adding more nanoparticles to a liquid makes it better at conducting heat, which helps the liquid transfer heat more efficiently. The findings indicate that, in all instances, isothermal obstacles lead to a more significant increase in the average Nusselt number compared to adiabatic obstacles.

Asymptotically consistent and computationally efficient modeling of short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations

This article proposes a novel computational modeling approach for short-ranged molecular interactions between curved slender fibers undergoing large 3D deformations, and gives a detailed overview how it fits into the framework of existing fiber or beam interaction models, either considering microscale molecular or macroscale contact effects. The direct evaluation of a molecular interaction potential between two general bodies in 3D space would require to integrate molecule densities over two 3D volumes, leading to a sixfold integral to be solved numerically. By exploiting the short-range nature of the considered class of interaction potentials as well as the fundamental kinematic assumption of undeformable fiber cross-sections, as typically applied in mechanical beam theories, a recently derived, closed-form analytical solution is applied for the interaction potential between a given section of the first fiber (slave beam) and the entire second fiber (master beam), whose geometry is linearly expanded at the point with smallest distance to the given slave beam section. This novel approach based on a pre-defined section–beam interaction potential (SBIP) requires only one single integration step along the slave beam length to be performed numerically. In addition to significant gains in computational efficiency, the total beam–beam interaction potential resulting from this approach is shown to exhibit an asymptotically consistent angular and distance scaling behavior. Critically for the numerical solution scheme, a regularization of the interaction potential in the zero-separation limit as well as the finite element discretization of the interacting fibers, modeled by the geometrically exact beam theory, are presented. In addition to elementary two-fiber systems, carefully chosen to verify accuracy and asymptotic consistence of the proposed SBIP approach, a potential practical application in form of adhesive nanofiber-grafted surfaces is studied. Involving a large number of helicoidal fibers undergoing large 3D deformations, arbitrary mutual fiber orientations as well as frequent local fiber pull-off and snap-into-contact events, this example demonstrates the robustness and computational efficiency of the new approach.

Full-field deformation reconstruction of beams using the inverse Finite Element Method: Application to thin-walled structures

This work presents a methodology for the real-time structural deformation reconstruction of solid or thin-walled prismatic beams using discrete strains. Based on the 1D inverse Finite Element Method (iFEM), the approach combines the beam kinematics of Timoshenko theory, strain–displacement relations, and the finite element discretisation framework to reconstruct the full-field deformations of beams with any general cross-section. Although applicable for any general class of beams, the present work investigates its experimental application specifically to conventionally and additively manufactured thin-walled beams discretised using various low- and high-order inverse beam elements. The results demonstrate the method’s accuracy and robustness, albeit influenced by the element discretisation scheme and the number of strain sensors used.

Propagation and complex morphology of hydraulic fractures in lamellar shales based on finite-discrete element modeling

We explore the controls of stress magnitude and orientation relative to bedding on the resulting morphology/topology of hydraulic fractures using a combined finite-discrete element method (FDEM). Behavior is shown conditioned by the ratio of principal stresses λ=σ3/σ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda ={\\sigma }_{3}/{\\sigma }_{1}$$\\end{document} and relative inclination of the bedding. When the lateral pressure coefficient (λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda$$\\end{document}) is less than 0.67, hydraulic fractures predominantly initiate as tensile fractures along the wellbore, aligning with the maximum principal stress direction. Conversely, for λ≥0.67\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\ge 0.67$$\\end{document}, shear cracks are favored to initiate for the minor stress difference, leading to a less predictable initiation and extension direction. Simultaneously, diminished stress differences correspond to elevated reservoir breakdown pressures, displaying a linear correlation with lateral pressure coefficients and little influenced by equivalent bedding orientation. Bedding plane orientation significantly impacts the mode and morphology of hydraulic fracture propagation. Bedding parallel to the direction of the minimum principal stress (σ3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\sigma }_{3}$$\\end{document}) favors layer-penetrating and bifurcated fractures, whereas inclined bedding facilitates the emergence of numerous steering-type and capture-type fractures. Especially at steeper inclinations (β=60∘\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta =60^\\circ$$\\end{document}), hydraulic fractures readily extend along the bedding surface, inducing macroscopic shear slip failure. Under high-stress disparities, the breakdown pressure exhibits greater sensitivity to bedding inclination, and its influence pattern aligns with the variations in tensile strength, typically reaching maximum and minimum values at bedding inclination angles of 0° and 60°, respectively.

One- and two-level Arrow–Hurwicz-type iterative algorithms for the stationary Smagorinsky model

In this article, based on finite element discretization, we propose one- and two-level Arrow–Hurwicz-type iterative algorithms for solving the steady-state Smagorinsky equations. The two-level Arrow–Hurwicz-type iterative algorithm involves solving a linearization Smagorinsky problem by the Arrow–Hurwicz-type iteration on a coarse mesh with mesh size H, and one Oseen-type linear problem on a fine mesh with mesh size h. Stability analysis and error estimation are provided. Compared with the classical Arrow–Hurwicz algorithm, the proposed algorithms have less constriction concerning parameter. Finally, we test some numerical experiments to illustrate the effectiveness of the proposed algorithms.

Superconvergence error analysis of linearized semi-implicit bilinear-constant SAV finite element method for the time-dependent Navier–Stokes equations

In this paper, based on the scalar auxiliary variable (SAV) approach, the superconvergence error analysis is investigated for the time-dependent Navier–Stokes equations. In which, an equivalent system of the Navier–Stokes equations with three variables and a fully-discrete scheme is developed with semi-implicit Euler discretization for the temporal direction and low-order bilinear-constant finite element discretization for the spatial direction, respectively. With the help of the high-precision estimations of the bilinear-constant finite element pair on the rectangular meshes, the superclose error estimates for velocity in H1-norm and pressure in L2-norm are obtained by treating the trilinear term carefully and skillfully. The global superconvergence results are also derived in terms of a simple and efficient interpolation post-processing technique. Finally, some numerical results are provided to demonstrate the correctness of the theoretical analysis.

Numerical analysis of a nonsmooth quasilinear elliptic control problem: I. Explicit second-order optimality conditions

In this paper, we derive explicit second-order necessary and sufficient optimality conditions of a local minimizer to an optimal control problem for a quasilinear second-order partial differential equation with a piecewise smooth but not differentiable nonlinearity in the leading term. The key argument rests on the analysis of level sets of the state. Specifically, we show that if a function vanishes on the boundary and its the gradient is different from zero on a level set, then this set decomposes into finitely many closed simple curves. Moreover, the level sets depend continuously on the functions defining these sets. We also prove the continuity of the integrals on the level sets. In particular, Green's first identity is shown to be applicable on an open set determined by two functions with nonvanishing gradients. In the second part to this paper, the explicit sufficient second-order conditions will be used to derive error estimates for a finite-element discretization of the control problem.

Numerical investigation of mode I fracture toughness anisotropy of deeply textured shale

The fracturing behavior of shale is controlled by the complex interaction between texture plane and rock matrix. This study develops a numerical model of semicircular bending of textured shale, integrating considerations of texture inclination and strength using the finite discrete element method, to examine the impact of shale anisotropy on damage patterns and fracture toughness. The results disclose the following insights: (1) Mode I fracture toughness anisotropy is significant, with fracture toughness showing a negative correlation with texture inclination and a positive correlation with texture strength. (2) Typical failure modes of SCB specimens encompass tensile failure across textured planes, tensile-shear mixed failure, tensile failure along texture planes, and tensile failure along the matrix. (3) For inclination angles ranging from 0° to 90°, cracks tend to deflect along the texture, leading to a fracture surface with high roughness. The fracture surface roughness was higher along the direction of maximum principal stress for the 0° and 90° specimens compared to the Divider specimens, while it was lower along the thickness direction compared to the Divider specimens. (4) The toughening mechanism of textured shales is categorized into three classes: separation along the texture, fracture path deflection, and texture cracking. Texture strength mainly influences texture cracking, while texture inclination primarily affects the deflection of crack paths and separation along the texture. (5) The criterion for predicting modified fracture toughness accurately predicts the anisotropic fracture toughness of shale compared to the conventional criterion for predicting fracture toughness.

Finite element method for a generalized constant delay diffusion equation

This paper considers the finite element method to solve a generalized constant delay diffusion equation. The regularity of the solution of the considered model is investigated, which is the first time to discover that the solution has non-uniform multi-singularity in time compared with Tan et al. (2022). To overcome the multi-singularity, a symmetrical graded mesh is used to devise the fully discrete finite element scheme for the considered problem based on L1 formula of the Caputo fractional derivative and fractional trapezoidal formula of the Riemann–Liouville fractional integral. Then we investigate the unconditional stability of this scheme. Next, the local truncation errors of the L1 formula and the fractional trapezoidal formula are analyzed in detail, especially the later one is discussed at the first time, under the multi-singularity of the solution and the symmetrical graded mesh. Using these error results, we obtain the convergence of the proposed numerical scheme. Finally, some numerical tests are provided to verify the obtained theoretical results.

Modeling the failure process of rock masses using a 3D nodal-based continuous-discontinuous deformation analysis method

A three-dimensional nodal-based continuous-discontinuous deformation analysis method (3D-NCDDAM) is developed in this study for modeling the failure process of rock masses. In the 3D-NCDDAM, four-node tetrahedral elements which can be automatally generated are used to discretize the problem domain. To reduce computational cost, and effectively model the failure process of rock masses at concerned regions, the problem domain is devided into two types of regions, namely, the “non-destructive region” and “possible destructive region”. The non-destructive region is mainly used to reduce the boundary effects on the numerical results, and is regarded transitional zone in which the failure of rock masses is not considered. For the possible destructive region, artificial joint elements are inserted, and the failure of these joint elements is used to represent fracture initiation and propagation in rock masses. Additionally, the contact potential which is initially proposed in finite-discrete element method (FDEM) is incorporated into the 3D-NCDDAM to simuate the opening and slipping of discontinuous structural planes involved in rock masses’ failure process. Note that the difficultly in determing contact types in the traditional 3D discontinuous deformation analysis (3D-DDA) method is avoided in the 3D-NCDDAM. The proposed 3D-NCDDAM can be treated as a hybrid method, which inherite both the advantage of DDA method and FDEM. To validate the 3D-NCDDAM, seven numerical examples are conducted. The numerical results indicte that the 3D-NCDDAM can effectively model the failure process of rock masses in both experimental scale and engineering scale.

Numerical investigation on the fracture and mechanical behaviors of marble containing two parallel fissures under triaxial compression conditions using FDEM

This paper aims to investigate the fracture and mechanical behaviors of fissured marble under triaxial compression. The two-dimensional models of marble with two parallel fissures and various rock bridge angles were established, and then the numerical simulations considering the effect of confining pressure were performed with the finite-discrete element method. The brittle-ductile transition and stress multi-peak characteristics are presented with the increase of the confining pressure. The characteristic strain, stress, and elastic strain energy exhibit various trends with the increment of the rock bridge angle, whilst increasing trend with the increment of the confining pressure. The cohesion presents the variation of first increasing and then decreasing with the increase of the rock bridge angle, and the friction angle presents the opposite trend to the cohesion. The proposed strength model can reflect the evolutions of the cohesion and the friction angle considering the effect of confining pressure and rock bridge angle. The elastic strain energy has the trends of nonlinear increase and fluctuation in the pre-peak and post-peak phases, respectively. The variation of kinetic energy is induced by the internal local fracture during the progressive failure. The failure pattern transforms from the tensile failure to the shear failure with the increase of the confining pressure. The coalescence behavior of cracks at the rock bridge can be classified into four modes.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

Coupled 3D thermal-hydraulic code development for performance assessment of spent nuclear fuel disposal system

As a solution to the problem of spent nuclear fuels (SNFs), the disposal of SNF has gained attention from nations using nuclear energy because of hazards posed to the ecosystem. Among many proposed solutions, the most promising method is to dispose of SNF in a deep geological repository (DGR) which utilizes the multi-barrier concept developed by Finland and Sweden. Here, a new fully-coupled Thermal-Hydraulic (TH) code HADES (High-level rAdionuclide Disposal Evaluation Simulator) is developed using the MOOSE framework. This new code suggests basic numerical tools, such as a non-linear solver and finite element discretization, to assess the safety performance of disposal systems. The new TH code considered various TH behavior using Richards' flow approach, assuming gas pressure is constant. The HADES showed promising results when it was compared to various TH codes validated from DECOVAELX-THMC projects. When the single-canister model was utilized to estimate the TH behavior of the Korean Reference disposal System, although it showed significant saturation reduction due to the evaporation of water, the temperature was maintained under the thermal criteria limit, which is 100 °C. In addition, the new code estimated temperature and degree of saturation of the multi-canisters model, considering two or three canisters, it showed a slightly lower temperature, 5 °C, than the single-canister model. From these results, the following are concluded: (1) the new TH code contribute to an additional integrity by estimating TH behavior of KRS; (2) however, due to limitations in single-canister simulation, it is recommended to use multi-canisters simulation to estimate TH behavior accurately. Therefore, this model is anticipated not only to help licensing applications and estimation of various multi-physics phenomena and multi-canister at the disposal site.

Numerical Investigation of Tree-Type Hydraulic Fracturing for Balanced Permeability Enhancement of Heterogenous Coal Seams Based on the Finite-Discrete Element Method Model.

Tree-type hydraulic fracturing (TTHF) is a new technology that can enhance the permeability of coal seams in a balanced manner and increase the coalbed methane production rate. However, the heterogeneity of coal seams is a major challenge in achieving balanced permeability enhancement by TTHF. Traditional methods based on digital image processing are difficult to apply in practice. To address these challenges, we proposed a 2D numerical model of coal seams based on the combined finite-discrete element method (FDEM). The elastic modulus of the coal seams obeys a Weibull distribution, and the coal heterogeneity was quantified by an index m. The effects on the fracture initiation pressure, the fracturing influence range, and displacements of TTHF were analyzed from four aspects, including the homogeneity index of coal, the arrangement angle of branch boreholes, the horizontal stress difference, and the injection rate of the fracturing fluid. The results show that TTHF has a significant effect on the balanced permeability enhancement in coal reservoirs, particularly with strong heterogeneity, and the best permeability enhancement for TTHF is achieved when the branch boreholes are arranged at 45°. The branch boreholes are prefabricated in advance to create a pressure relief area around the injection point, and the hydraulic fracture propagation is affected by the horizontal stress difference only when the fracturing influence range exceeds this area. When the horizontal stress difference increases from 0 to 4 MPa, its fracture initiation pressure increases from 8.93 to 10.86 MPa, with an increase of 21.61%. In addition, the initial stage of fluid injection was found to be crucial for achieving balanced permeability enhancement in TTHF, and a higher injection rate can expand the fracturing influence range. The numerical model has profound implications for the field application of TTHF technology.

A 3D discrete model for soil desiccation cracking in consideration of moisture diffusion

Soil desiccation cracking is a common phenomenon on the earth surface. Numerical modeling is an effective approach to study the desiccation cracking mechanism of soil. However, no three-dimensional (3D) model in the literature can consider the influence of desiccation cracking on moisture migration. To fill this gap, this work develops a novel 3D moisture diffusion discrete model that is capable of dynamically assessing the effect of cracking on moisture diffusion and allowing moisture to be discontinuous on both sides of the cracks. Then, the parametric analysis of the moisture exchange coefficient in the 3D moisture diffusion discrete model is carried out for moisture diffusion in continuous media, and the selection criterion of the moisture exchange coefficient for the unbroken cohesive element is given. Subsequently, an example of moisture migration in a medium with one crack is provided to illustrate the crack hindering effect on moisture migration. Finally, combining the 3D moisture diffusion discrete model with the finite-discrete element method (FDEM), the moisture diffusion-fracture coupling model is built to study the desiccation cracking in a strip soil and the crack pattern of a rectangular soil. The evolution of crack area and volume with moisture content is quantitatively analyzed. The modeling number and average width of cracks in the strip soil show a good consistency with the experimental results, and the crack pattern of the rectangular soil matches well with the existing numerical results, validating the coupled moisture diffusion-fracture model. Additionally, the parametric study of soil desiccation cracking is performed. The developed model offers a powerful tool for exploring soil desiccation cracking.

Capacity fading and inter/intra-granular fracture of polycrystalline cathode particle induced by charge/discharge cycles: A numerical study

The capacity fading of Lithium-ion batteries (LIBs) containing polycrystalline cathode materials can be attributed to the degradation of secondary particles caused by inter- and intra-granular fractures occurring during charging and discharging cycles. In this work, a diffusion-induced stress (DIS) analysis method based on the finite-discrete element method (FDEM) is developed for studying the fracture behaviors of polycrystalline cathodes. The analysis of lithiation and delithiation processes shows that C-rate and interfacial fracture toughness have significant effects on inter-granular crack morphologies. Our method enables precise analysis of the competition between intra- and inter-granular cracks and the effect of crack-impeded diffusion on crack morphologies. We compared numerical simulation results for scenarios where fracture toughness changes with the number of cycles, and demonstrated that the decreasing fracture toughness with cycle number is an important factor contributing to the slow crack growth. Additionally, we established the relationship between crack length and capacity loss, which agreed well with experimental results.

FDEM analysis of deep rock mass failure and its impact on horizontal bearing capacity in offshore wind turbine piles

Rock-socketed single piles are integral components of offshore wind turbine foundations in rock-based sea areas. Traditional numerical models have been employed to analyze the failure modes and lateral interactions between piles and the rock medium. However, these do not simulate the decline in the horizontal bearing capacity due to rock breakage, posing potential safety risks. In this study, the combined finite-discrete element method (FDEM) was used to simulate the deformation process of submarine rock masses transitioning from an intact to a fractured state. A notable innovation in the proposed approach is the consideration of the effects of the rock mass fracture and pile-rock cohesion on the load-bearing behavior of single-pile foundations. The FDEM model was validated through full-scale lateral load tests. The results showed a distinctive sequence in failure mechanisms, with the tensile failure of the rock mass around the pile at significant depths preceding the compression failure. Local failure at the contact interface significantly reduced the horizontal bearing capacity—a phenomenon that is inadequately represented by conventional finite element models. Furthermore, the cohesion element embedding method was introduced and used to simulate the influence of the tensile properties inherent in grouting materials on the failure strength and mode of the pile-rock interface. This study highlighted the importance of considering the nuanced effects of rock failure in the design of horizontally loaded piles. Failure to do so could lead to nonconservative designs and compromise the safety and efficacy of offshore wind turbine foundations. By pushing the boundaries of numerical modeling with the FDEM, the study aims to provide a more accurate and reliable framework for the design and assessment of rock-socketed single piles for offshore wind energy applications.