Gauss Points Research Articles

Physics-informed neural networks for V-notch stress intensity factor calculation

This paper proposes a physics-informed neural networks (PINNs) based approach for elastic structures with a V-notch, by which the displacement field, stress field as well as the V-notch stress intensity factor (NSIF) can be obtained through artificial neural networks. A PINN model is established for V-notch structures, integrating physical information into a deep neural network to ensure adherence to physical laws while fitting observational data. Subsequently, an adaptive local sampling strategy for V-notch structures is adopted, generating locally dense Gaussian points sampling around regions of stress concentration. Based on this, a sequential PINNs approach for V-notch structures is then established to calculate the NSIF for V-notch structures with arbitrary notch angles. Finally, the effectiveness of the proposed method is validated through three numerical examples. The results demonstrate the method can accurately predict the NSIFs for V-notch structures across a spectrum of opening angles. Compared to the traditional data-driven method, the proposed method is able to more effectively compute the NSIF of V-notch structures due to the integration of physical information and observational data.

Three-dimensional buckling analysis of stiffened plates with complex geometries using energy element method

A novel numerical method, energy element method (EEM), is proposed for the three-dimensional (3D) buckling analysis of stiffened plates with complex geometries. The problem is formulated in a cuboidal domain, and any complex geometric stiffened plate is modeled by assigning cutouts within the cuboidal domain. The stiffened plate is considered as an energy body and is discretized using Gauss points with variable stiffness properties to simulate its energy distribution. Incorporating the extended interval integration, Gauss quadrature, variable stiffness properties, and Chebyshev polynomials, the strain energy of stiffened plates with complex geometries can be numerically simulated by putting the stiffness and thickness of Gauss points in the cutouts to zero in the cuboidal domain. Using the principle of minimum potential energy and Ritz solution procedure, the deformation and buckling behaviors of stiffened plates with complex geometries can be captured. As a result of the new formulations in EEM, new standard energy functionals and solving procedures have been developed. In addition, Gauss points are generated within the energy elements accounting for the geometric boundaries of the stiffened plate, which are characterized by level set functions. EEM is employed to investigate complex-shaped stiffened plates with straight or curvilinear stiffeners, and the results are compared to those obtained using FEM or mesh-free method. The precision, generalization, and stability of EEM are demonstrated.

On sequences of convex records in the plane

Abstract Convex records have an appealing purely geometric definition. In a sequence of d-dimensional data points, the nth point is a convex record if it lies outside the convex hull of all preceding points. We specifically focus on the bivariate (i.e. two-dimensional) setting. For iid (independent and identically distributed) points, we establish an identity relating the mean number ⟨ R n ⟩ of convex records up to time n to the mean number ⟨ N n ⟩ of vertices in the convex hull of the first n points. By combining this identity with extensive numerical simulations, we provide a comprehensive overview of the statistics of convex records for various examples of iid data points in the plane: uniform points in the square and in the disk, Gaussian points and points with an isotropic power-law distribution. In all these cases, the mean values and variances of Nn and Rn grow proportionally to each other, resulting in the finite limit Fano factors FN and FR . We also consider planar random walks, i.e. sequences of points with iid increments. For both the Pearson walk in the continuum and the Pólya walk on a lattice, we characterise the growth of the mean number ⟨ R n ⟩ of convex records and demonstrate that the ratio R n / ⟨ R n ⟩ keeps fluctuating with a universal limit distribution.

A subspace‐adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models

AbstractThis article is concerned with quadrature/cubature rules able to deal with multiple subspaces of functions, in such a way that the integration points are common for all the subspaces, yet the (nonnegative) weights are tailored to each specific subspace. These subspace‐adaptive weights cubature rules can be used to accelerate computational mechanics applications requiring efficiently evaluating spatial integrals whose integrand function dynamically switches between multiple pre‐computed subspaces. One of such applications is local hyperreduced‐order modeling (HROM), in which the solution manifold is approximately represented as a collection of basis matrices, each basis matrix corresponding to a different region in parameter space. The proposed optimization framework is discrete in terms of the location of the integration points, in the sense that such points are selected among the Gauss points of a given finite element mesh, and the target subspaces of functions are represented by orthogonal basis matrices constructed from the values of the functions at such Gauss points, using the singular value decomposition (SVD). This discrete framework allows us to treat also problems in which the integrals are approximated as a weighted sum of the contribution of each finite element, as in the energy‐conserving sampling and weighting method of C. Farhat and co‐workers. Two distinct solution strategies are examined. The first one is a greedy strategy based on an enhanced version of the empirical cubature method (ECM) developed by the authors elsewhere (we call it the subspace‐adaptive weights ECM, SAW‐ECM for short), while the second method is based on a convexification of the cubature problem so that it can be addressed by linear programming algorithms. We show in a toy problem involving integration of polynomial functions that the SAW‐ECM clearly outperforms the other method both in terms of computational cost and optimality. On the other hand, we illustrate the performance of the SAW‐ECM in the construction of a local HROMs in a highly nonlinear equilibrium problem (large strains regime). We demonstrate that, provided that the subspace‐transition errors are negligible, the error associated to hyperreduction using adaptive weights can be controlled by the truncation tolerances of the SVDs used for determining the basis matrices. We also show that the number of integration points decreases notably as the number of subspaces increases, and that, in the limiting case of using as many subspaces as snapshots, the SAW‐ECM delivers rules with a number of integration points only dependent on the intrinsic dimensionality of the solution manifold and the degree of overlapping required to avoid subspace‐transition errors. The Python source codes of the proposed SAW‐ECM are openly accessible in the public repository https://github.com/Rbravo555/localECM.

Energy Element Method for Three-Dimensional Vibration Analysis of Stiffened Plates with Complex Geometries

A novel numerical method, the energy element method, is proposed for the three-dimensional vibration analysis of stiffened plates with complex geometries. The problem is modeled in a cuboidal domain, and a cuboidal energy element is developed for the simulation of the structural components. To simulate the energy distribution of the structure, stiffened plates are treated as discrete energy systems, where variable stiffness is used to characterize their energy distribution in a cuboidal domain and a global admissible function is used to approximate their vibrational behavior. With extended interval integral, Gauss quadrature, variable stiffness, and Legendre polynomials used for numerical integration in the cuboidal domain, the cuboidal energy element can offer sufficient precision with sufficient Gauss points for simulating the strain energy of stiffened plates with complex geometries. The Gauss–Legendre quadrature will provide accurate integration results if the integral domain is cuboidal. Otherwise, small cuboidal energy elements are generated with denser Gauss points to capture the geometric boundaries. As the numerical model is constructed on a standard geometric domain, all energy functionals and computational procedures are standard. Three-dimensional vibration problems are investigated for variously shaped stiffened plates with straight or curvilinear stiffeners. The present results are compared with analytical, numerical, and experimental results published in the literature.

Implementation of the Gauss-Kronrod Quadrature Method (G7, K15) on 2D Gravity Anomaly Modeling in Basins with a Polynomial Variation of Density Distribution with Depth

Forward modeling of 2D gravity anomalies, considering density contrasts that vary polynomially with depth, was performed to examine basin structures. This process involved two main stages: deriving analytical formulas and executing numerical integration. The Gauss-Kronrod Quadrature Method, utilizing 7 Gauss points and 15 Kronrod points, was employed to precisely compute these integrals. Initial modeling applied to theoretical basement scenarios with fixed density contrasts showed gravity anomalies that accurately reflected the curvature of the basement. To validate the approach, it was then applied to real-world cases including the Sebastian Vizcaino Basin, San Jacinto Graben, and Sayula Basin. By incorporating suitable density contrasts, modeling lengths, and basement curvature shapes, the results revealed that both fixed-density and depth-variable density models produced gravity anomalies with patterns consistent with the actual basement curvature. These findings validate the modeling technique’s effectiveness in representing real geological features accurately. The study confirms that the Gauss-Kronrod Quadrature Method (G7, K15) is robust for analyzing 2D gravity anomalies, providing a reliable tool for understanding the influence of varying density contrasts on gravity responses.

An assumed enhanced strain finite element framework for tensile fracturing processes with dual-mechanism failure in transversely isotropic rocks

We present an assumed enhanced strain finite element framework for the simulation of tensile fracturing processes in transversely isotropic rocks. Fractures along the weak bedding planes and through the anisotropic rock matrix are treated with distinct enrichment, and a recently proposed dual-mechanism tensile failure criterion for transversely isotropic rocks is adopted to determine crack initiation for the two failure modes. The cohesive crack model is adopted to characterize the response of embedded cracks. As for the numerical implementation of the proposed framework, both algorithms for the update of local history variables at Gauss points and of the global finite element system are derived. Four boundary-value problem simulations are carried out with the proposed framework, including uniaxial tension tests of Argillite, pre-notched square loaded in tension, three-point bending tests on Longmaxi shale, and simulations of tensile cracks induced by a strip load around a tunnel in transversely isotropic rocks. Simulation results reveal that the proposed framework can properly capture the tensile strength anisotropy and the anisotropic evolution of tensile cracks in transversely isotropic rocks.

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

Improved Gaussian processes linear JPDA filter for multiple extended targets tracking in dense clutter

To track multiple extended targets in dense clutter, an improved Gaussian processes linear joint probabilistic data association (IGP-LJPDA) method is proposed. This method consists of two stages. In the first stage, measurements are associated with targets, and pseudo-measurements are constructed. By integrating linear JPDA with Gaussian processes, probabilities for each measurement's assignment to each target are determined. Following this, a novel pseudo-measurement model is devised by combining the marginal probability of each target with measurements belonging to their respective basic point neighborhood, enabling more efficient state updates. In the second stage, pseudo-measurements are associated with the contour, and states of extended targets are updated accordingly. By associating pseudo-measurements with Gaussian processes basic points of each target, the algorithm achieves globally optimal association. The proposed algorithm demonstrates a notable improvement in Intersection over Union (IOU) metrics in environments with significantly higher clutter rates compared to existing methods.

Deblur-GS: 3D Gaussian Splatting from Camera Motion Blurred Images

Novel view synthesis has undergone a revolution thanks to the radiance field method. The introduction of 3D Gaussian splatting (3DGS) has successfully addressed the issues of prolonged training times and slow rendering speeds associated with the Neural Radiance Field (NeRF), all while preserving the quality of reconstructions. However, 3DGS remains heavily reliant on the quality of input images and their initial camera pose initialization. In cases where input images are blurred, the reconstruction results suffer from blurriness and artifacts. In this paper, we propose the Deblur-GS method for reconstructing 3D Gaussian points to create a sharp radiance field from a camera motion blurred image set. We model the problem of motion blur as a joint optimization challenge involving camera trajectory estimation and time sampling. We cohesively optimize the parameters of the Gaussian points and the camera trajectory during the shutter time. Deblur-GS consistently achieves superior performance and rendering quality when compared to previous methods, as demonstrated in evaluations conducted on both synthetic and real datasets.

Dynamic response of H-FGS plates integrated with Pasternak foundation subjected to harmonic loading in the thermal environment using Chebyshev-FEM

The main goal of this article is to introduce a finite element method (FEM) based on Chebyshev polynomials, referred to as Chebyshev-FEM, to investigate the dynamic response of functionally graded sandwich (FGS) plates with a honeycomb core and two functionally graded (FG) skin layers, denoted as H-FGS plates integrated with Pasternak foundation (PF) subjected to harmonic loading (HL) within a thermal environment. Chebyshev polynomials are a series of orthogonal polynomials defined recursively. The value of the polynomials belongs to the range [–1] and vanishes at Gauss points. The equation of motion for H-FGS plates is derived using Hamilton’s principle. Comparative examples are implemented to validate the accuracy and performance of the proposed method. Additionally, the effect of input parameters on both free and forced vibrations of H-FGS plates integrated with PF subjected to HL within a thermal environment is studied in detail.

Buckling optimization of variable-stiffness composite plates with two circular holes using discrete Ritz method and potential flow

Potential flow around two equal-radius cylinders is derived analytically and applied to generate the curvilinear fiber path of variable-stiffness composite (VSC) plates with two circular holes. As complex variable theory and conformal mapping are used to generate the potential flow around two equal-radius cylinders, the location and size of the two equal-radius circular holes are arbitrary. By changing the angle of incoming flow, the global fiber angle of a variable-stiffness lamina can be simulated and the local fiber orientation angle at any point is determined by the global potential flow field. Buckling performance of variously-shaped VSC plates with two circular holes are investigated via a novel numerical method − discrete Ritz method (DRM). DRM combines the extended interval integral, Gauss quadrature, and variable stiffness within a rectangular domain and builds a discrete energy system to simulate the plate, which allows the geometric boundaries of the plate to vary. The strain energy of a plate is modeled in the rectangular domain, discretized using Gauss points, and is characterized by variable stiffness, which characterizes both the material distribution and plate geometry simultaneously. After that, a three-dimensional sampling optimization (3DSO) method is adopted to optimize the curvilinear fiber configurations of VSC plates, and its buckling performances are compared with those of constant stiffness composites (CSC) with optimal straight fibers. Significant improvements on load-carrying capacity can be achieved compared to straight ones, demonstrating that using potential flow is one of the most efficient way to generate curvilinear optimal fiber path with maximum load-carrying capacity for VSC plates with material discontinuity. Moreover, DRM exhibits good precision and stability for buckling analysis of VSC plates with complex geometries.

Multiscale modelling of composite laminates with voids through the direct FE2 method

A concurrent multiscale method, namely the direct FE2, is developed to analyse the mechanical response of fibre-reinforced composites with several void fractions in a nested manner, one at the macro and another one at the microscale. The FE2 is developed through the finite element (FE) mesh at the macro scale, wherein there is a representative volume element (RVE) at every Gauss point where the strains and stresses are evaluated. The RVEs, located at macroscopic FE mesh, with linear boundary conditions applied to the boundary nodes, are linked to macro-scale nodes via Multi-Point Constraints (MPCs). This procedure enables solving a single equilibrium problem, with boundary conditions and loadings at the macro scale and provides data for all defined RVEs. The methodology is applied to a composite laminate under transverse tension. Initially, the method is evaluated to compare the results from the FE2 with those from analyses where the entire domain is discretised (conventional FE full-scale modelling). The stresses obtained from the FE2 method match traditional FE analyses but utilise a significantly lower number of finite elements, resulting in substantial computational cost savings. It follows a parametric study of void distribution effects within the RVE. Afterwards, the stress distribution within the RVE is assessed and then failure of the composites is predicted at the macro scale through the stresses and strains concurrently calculated at the micro-scale.

Variability and loss of uniqueness of numerical solutions in FEM×DEM modeling with second gradient enhancement

AbstractIn the last decade, a new multi‐scale FEM×DEM approach has been developed using Finite Element Method (FEM) coupled with Discrete Element Method (DEM) as a constitutive law to account for the specificities of the mechanical behavior of granular materials. In FEM×DEM model, a DEM calculation is performed on a particle assembly (volume element—VE) at each Gauss point. Recent publications have demonstrated that FEM×DEM approach naturally captures the discrete and anisotropic nature of granular materials. Despite its advantages, FEM×DEM with classical FEM, suffers from mesh dependency, especially when material enters softening phase and exhibits strain localization. To overcome this limitation, FEM×DEM model has been enriched by incorporating a local second gradient model. Nevertheless, the existence of multiple possible solutions is observed. In this paper, we study the variability and loss of uniqueness of numerical solutions to a boundary value problem. Different VEs with equivalent mechanical properties are generated and used to model the pressuremeter tests by means of FEM×DEM. The modeling results show a great variability of the numerical results, both in shape of borehole and in different modes of shear bands. For the same VE, the loss of uniqueness of numerical solutions is evidenced by a slight modification of loading history at the level of the internal pressure applied to the borehole. Finally, we show that when a certain heterogeneity is introduced by using different VEs within the same BVP, even if the uniqueness of the solution is not guaranteed, the set of possible solutions seems more restrained.

Efficient numerical implementation of limit equilibrium method for stability analysis of unsaturated soil slopes using Gaussian integral

Unsaturated soil is widely distributed around the world but less considered in design due to the absence of a convenient analysis method in practice. The Morgenstern–Price (MP) method incorporating the extended Mohr–Coulomb shear strength equation provides a reliable approach to evaluate slope stability in such conditions. However, this method is time-consuming due to the need for a tedious trial-and-error process in determining the scaling factor, which involves complex iterations during each trial. Furthermore, since the relatively complicated nature of unsaturated soil, a dense slice division is necessary to obtain reliable results, making the analysis even more cumbersome. In this paper, an improved MP method for unsaturated soil slope stability analysis is presented, which eliminates the need for a dense slice mesh by employing only a few strategically placed Gauss points along the slip surface. Moreover, the trial-and-error process for determining the scaling factor with the corresponding complex iterations is replaced by an efficient search algorithm with a more concise iteration process, resulting in a more convenient implementation of the proposed method. Extensive examples are provided to validate the effectiveness of the proposed improved MP method, indicating its potential as an accurate and efficient analysis method for unsaturated soil slopes in practical application and relative study involving repetitive analyses.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

A FE2 shell model with periodic boundary conditions for thin and thick shells

AbstractIn this article a FE2 shell model for thin and thick shells within a first order homogenization scheme is presented. A variational formulation for the two‐scale boundary value problem and the associated finite element formulation is developed. Constraints with 5 or 9 Lagrange parameters are derived which eliminate both rigid body movements and dependencies of the shear stiffness on the size of the representative volume elements (RVEs). At the bottom and top surface of the RVEs which extend through the total thickness of the shell stress boundary conditions are present. The periodic boundary conditions at the lateral surfaces are applied in such a way that particular membrane, bending and shear modes are not restrained. This is shown by means of a homogeneous RVE. The first of all linear formulation is extended to finite strain problems introducing transformation relations for the stress resultants and the material matrix. The transformations are performed at the Gauss points on macro level. Several boundary value problems including large deformations, stability and inelasticity are computed and compared with 3D reference solutions.

One-dimensional parallel forward modeling for geophysical electromagnetic fields excited by sagging overhead transmission lines

Geophysicists often regard the 50/60 Hz electromagnetic (EM) waves and their harmonics, excited by overhead transmission lines, as noise that interferes with geophysical prospecting. However, some researchers view them as effective signals for geophysical exploration. Forward modeling can be used to study the characteristics of EM fields generated by overhead transmission lines. It can help in either eliminating or utilizing power-frequency EM fields. Nevertheless, current research on the forward modeling of EM fields excited by transmission lines neglects the sagging shape and direction of the actual transmission lines, simplifying them into infinitely long straight lines. In this research, we develop a one-dimensional forward modeling algorithm specifically designed for overhead transmission lines, which can accurately calculate the EM fields excited by realistic overhead transmission lines considering the three-phase system, design parameters, sag, and orientation of the transmission lines. To construct an accurate transmission line model, we represent the sagging shape of a single transmission line within a span by a catenary equation. We then derive analytical expressions for the EM fields excited by sagging overhead transmission lines. During forward modeling, an appropriate number of Gauss points must be selected to balance accuracy and speed. The optimal number of Gauss points is determined based on the distance between the observation point and the transmission line source, as well as the type of fields being calculated. OpenMP parallelization is used to improve computational speed. The forward modeling results reveal significant differences between the forward responses of a single infinitely long straight line and realistic transmission lines. This highlights the importance of fully incorporating the three-phase system, design parameters, and geometric shapes of transmission lines in the forward modeling process. Moreover, the inversion test demonstrates the potential utility of EM fields radiated by transmission lines for geophysical exploration.

A multi-domain hybrid spectral collocation method for nonlinear Volterra integral equations with weakly singular kernel

In this paper, we propose a multi-domain hybrid spectral collocation method for the nonlinear Volterra integral equations (VIEs), whose solutions exhibit weak singularities at the endpoint t=0. In order to efficiently approximate the weakly singular solutions, we divide the interval [0,T] into N subintervals and use the Gauss points of the generalized log orthogonal functions as the collocation points to approximate the weakly singular integral term in the first subinterval. In the remaining subintervals, we use Legendre and Jacobi Gauss points as the collocation points to approximate the corresponding integral terms. We also provide a rigorous hp-version convergence analysis for the hybrid spectral collocation method under L2-norm. A series of examples demonstrate our method is particularly suitable for solutions that have weak singularities at one endpoint.

Slope Stability Analysis Based on the Explicit Smoothed Particle Finite Element Method

A landslide is a common natural disaster that causes environmental damage, casualties and economic losses, which seriously affects the sustainable development of society. In geomechanics, it is one of the largest deformation problems. Herein, the GPU-accelerated explicit smoothed particle finite element method (eSPFEM) for large deformation analysis in geomechanics was developed on the CUDA platform based on high-performance computing using a self-designed eSPFEM program code. The eSPFEM combines the strain smoothing nodal integration techniques found in the particle finite element method (PFEM) framework, which allows for the use of low-order triangular elements without volume locking and avoids frequent information transfer and mapping errors between Gaussian points and particles in PFEM. A numerical simulation of slope instability using the eSPFEM and based on a strength reduction technique was conducted using various examples, including a cohesive homogeneous slope, a non-cohesive homogeneous slope, a non-homogeneous slope and a slope with a thin soft band. The calculation results show that the eSPFEM can be applied to slope stability analysis under different working conditions, simulating the entire process of slope instability initiation, sliding and reaccumulation, and obtaining reliable FOS values. A numerical simulation was conducted to analyse a landslide that occurred in the Zhangjiazhuang tunnel on the Lanzhou–Xinjiang high-speed railway line on 18 January 2016. A natural unsaturated soil slope, a soil slope with a high moisture content and a soil slope with a high moisture content subjected to an earthquake were analysed. The findings of this study are in good agreement with the actual slope failure conditions. The primary triggers identified for the landslide were heavy rainfall and earthquakes. The verification results indicate that the eSPFEM can effectively simulate an actual landslide case, showcasing high accuracy and applicability in simulating the large deformation behaviour of landslides.