Smoothed Particle Finite Element Method Research Articles

Modeling the large deformation failure behavior of unsaturated porous media with a two-phase fully-coupled smoothed particle finite element method

In this paper, a computational framework based on the Smoothed Particle Finite Element Method is developed to study the coupled seepage-deformation process in unsaturated porous media. Governing equations are derived from the balance laws of solid and fluid phases considering partial saturation effects in porous media. Moreover, an hourglass control method is implemented to avoid the rank-deficiency issue in SPFEM and the moving least squares approximation technique (MLS) is implemented to eliminate the pore pressure oscillations when the low-order triangle element is used. The proposed coupled SPFEM formulation is validated through four elastic examples and one elasto-plastic example. Good agreement with the numerical or analytical results reported in the literature is obtained. Further, the rainfall-induced slope failure is studied, in which a suction-dependent elasto-plastic Mohr–Coulomb model is adopted to take account of the suction effect in unsaturated soil. The evolution of the suction and soil deformation during the rainfall period and the whole slope failure process are obtained. It is demonstrated that the proposed method is a promising tool in numerical investigations of both the triggering mechanisms and post-failure behavior of the rainfall-induced slope failure.

Stabilizing nodal integration in dynamic smoothed particle finite element method: A simple and efficient algorithm

The Smoothed Particle Finite Element Method (SPFEM) is a powerful numerical tool for the modelling of large deformation problems in the field of geomechanics. However, the standard SPFEM with nodal integration suffers from the rank deficiency problem due to under-integration. In this study, an hourglass-control based stabilization scheme is presented to overcome the rank deficiency in standard SPFEM with nodal integration. By introducing the hourglass-control based correction forces, the spurious low-energy models in standard SPFEM with nodal integration can be effectively suppressed. The presented stabilization scheme is simple and computationally efficient compared to the existing strain gradient-based stabilization approaches in SPFEM. The proposed hourglass-control based stabilized SPFEM is firstly validated by means of three benchmark examples and the results show that the stabilization scheme based on the hourglass-control can eliminate the rank deficiency instability in standard SPFEM and free of volumetric locking. Finally, two numerical examples related to the cohesive soil flow and the progressive failure of sensitive clay slopes are studied to highlight the robustness of the proposed stabilized SPFEM in solving large deformation geotechnical problems.

ESPFEM2D: A MATLAB 2D explicit smoothed particle finite element method code for geotechnical large deformation analysis

ESPFEM2D: A MATLAB 2D explicit smoothed particle finite element method code for geotechnical large deformation analysis

A hydro-mechanical coupled contact method for two-phase geotechnical large deformation problems within the SNS-PFEM framework

This paper proposes a novel numerical algorithm to solve frictional contact problems in two-phase porous media, accounting for the large deformation and interface fluid flow, within the framework of the Stable Node-based Smoothed Particle Finite Element Method (SNS-PFEM). The proposed approach builds on our previous work that was only designed for single-phase large deformation contact problems. Within this work, the hydro-mechanical coupling system is discretized using equal-order interpolation, with a polynomial pressure projection method to eliminate the pore pressure instability. Surface-to-surface discretization is then applied to formulate the contact constraint of the two-phase porous medium. Using the dual Lagrange multiplier approach, a coordinate transformation is introduced to enforce the contact constraint of the solid phase. This transformation is subsequently utilized to develop a new interface fluid element, which serves as a tool to model the fluid exchange behaviour between porous media and contact gaps. Different from existing interface fluid elements, the proposed element is constructed based on the contact algorithm, thereby enabling it to handle non-conforming interfaces and tackle challenges associated with large sliding problems. Furthermore, the proposed method can maintain a well-conditioned system matrix throughout the computation process. These distinctive attributes significantly enhance the flexibility of numerical modelling. Finally, the accuracy and performance of our method are verified through four numerical examples. The results show that the proposed method can effectively simulate various geotechnical problems, including multi-layer soil consolidation, fluid-soil-structure interaction, and geological interface-controlled slope failure.

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.

Circumventing volumetric locking in stabilized smoothed particle finite element method and its application to dynamic large deformation problems

AbstractThe smoothed particle finite element method (SPFEM) is an effective framework for large deformation analysis. The original SPFEM possesses the rank deficiency issue due to the direct nodal integration technique, which can be overcome by incorporating the strain gradient stabilization method. However, an extra volumetric locking due to the strain gradient stabilization term has been found. In this study, we propose a simple and efficient approach, the B‐dev approach, which only considers the deviatoric part of the smoothed strain gradient for stabilization to overcome the volumetric locking issue. First, the correctness of the stabilized SPFEM method is verified by an elastic cantilever beam vibration problem without considering incompressibility. Then, through the example of the strip footing penetration problem, the volumetric locking in the stabilized SPFEM is demonstrated and the capability of the proposed B‐dev approach in terms of overcoming the volumetric locking is verified. Furthermore, the stabilized SPFEM with B‐dev approach is applied to two types of elastoplastic problems in geotechnical engineering. All numerical results illustrate that the proposed approach can improve the performance of the stabilized SPFEM in dealing with incompressible and large deformation problems in geomechanics.

A dynamic SNS-PFEM with generalized-α method for hydro-mechanical coupled geotechnical problems

The node-based smoothed particle finite element method (NS-PFEM) is computationally efficient with pure linear interpolation, while some intrinsic defects hinder its application in dynamic and hydro-mechanical coupled analysis. This study proposes a dynamic implicit stable NS-PFEM (SNS-PFEM) with some features: (1) the subdomain-based stress point stabilization is applied to overcome the spurious non-zero energy mode of NS-PFEM; (2) the polynomial pressure projection is implemented into the dynamic SNS-PFEM to guarantee the pore pressure stabilization; (3) the implicit time marching scheme of generalized-α method to integrate the stabilized equations is derived, which is error-controllable, unconditionally stable regardless of time step and permeability; and (4) an updated Lagrangian remeshing scheme is implemented to deal with the large deformation. The proposed implicit SNS-PFEM is first validated by several benchmark examples, and then applied to analyse the seabed under wave loading, the cutting slope experiment, and the slope failure under seismic loading. All results demonstrate the proposed method is stable and accurate in modelling the hydro-mechanical coupled dynamic problems under small and large deformation with nonlinear soil constitutive models and different drainage conditions.

A temporal stable smoothed particle finite element method for large deformation problems in geomechanics

In this study, a stabilized smoothed particle finite element method (SPFEM) for modeling nonlinear, large deformation dynamic problems in geomechanics is presented. The stabilization is introduced to the weak form formulations through a strain gradient of the smoothing cell, which helps eliminate the spurious zero-energy modes in SPFEM and achieve a temporal stable solution. Compared to the existing strain gradient stabilization approaches in SPFEM, the proposed stabilized SPFEM approach is computationally less intensive, free of artificial penalty parameters and free of volumetric locking. The performance of the proposed stabilized SPFEM is demonstrated by the studies of two numerical benchmark examples, the results of which show excellent agreement with the analytical reference solutions. This stabilization technique is also found to help improve the computation accuracy in terms of displacement and strain energy in comparison with results from the standard SPFEM. In addition, the proposed stabilized SPFEM is found to be temporal stable and free of volumetric locking. Finally, two numerical applications of the stabilized SPFEM to the large deformation soil flow and slope failure are discussed.

A kinetic energy-based failure criterion for defining slope stability by PFEM strength reduction

From the perspective of energy, the change of kinetic energy is rather obvious in the process of slope from steady state to failure. To numerically define the factor of safety (FOS) of the slope, the strength reduction method (SRM) is coupled with the two-dimensional explicit smoothed particle finite element method (eSPFEM), considered kinetic energy, to analyze slope stability. Through the failure analysis of slope, a new failure criterion based on the change of occurrence time of peak kinetic energy is proposed in this paper. Its applicability and accuracy are verified by two homogeneous slopes and two inhomogeneous slopes. The results show that the evolution of kinetic energy in the slope can intuitively and effectively reveal the sudden change of occurrence time of peak kinetic energy, leading to objectively determining the FOS of the slope. The proposed failure criterion regarding the characteristics of kinetic energy evolution could be further utilized in practical engineering to determine a more accurate factor of safety of the slope.

Stabilized smoothed particle finite element method for coupled large deformation problems in geotechnics

Stabilized smoothed particle finite element method for coupled large deformation problems in geotechnics

Numerical Analysis of an Explicit Smoothed Particle Finite Element Method on Shallow Vegetated Slope Stability with Different Root Architectures

Planting vegetation is an environmentally friendly method for reducing landslides. Current vegetated slope analysis fails to consider the influence of different root architectures, and the accuracy and effectiveness of the numerical simulations need to be improved. In this study, an explicit smoothed particle finite element method (eSPFEM) was used to evaluate slope stability under the influence of vegetation roots. The Mohr–Coulomb constitutive model was extended by incorporating apparent root cohesion into the shear strength of the soil. The slope factors of safety (FOS) of four root architectures (uniform, triangular, parabolic, and exponential) for various planting distances, root depths, slope angles, and planting locations were calculated using the shear strength reduction technique with a kinetic energy-based criterion. The results indicated that the higher the planting density, the stronger the reinforcement effect of the roots on the slope. With increasing root depth, the FOS value first decreased and then increased. The FOS value decreased with an increase in slope angle. Planting on the entire ground surface had the best improvement effect on the slope stability, followed by planting vegetation with a uniform root architecture in the upper slope region or planting vegetation with triangular or exponential root architecture on the slope’s toe. Our findings are expected to deepen our understanding of the contributions of different root architectures to vegetated slope protection and guide the selection of vegetation species and planting locations.

Overcoming volumetric locking in stable node‐based smoothed particle finite element method with cubic bubble function and selective integration

AbstractThe stable node‐based smoothed particle finite element method (SNS‐PFEM) reduces spatial numerical oscillation from direct nodal integration in NS‐PFEM but leads to a severe volumetric locking effect when modeling nearly incompressible materials‐related boundary value problems. This study proposes an improved locking‐free SNS‐PFEM to investigate the performance of the bubble function and selective integration scheme in circumventing volumetric locking. Three locking‐free variants of SNS‐PFEM: (1) SNS‐PFEM with a cubic bubble function (bSNS‐PFEM), (2) SNS‐PFEM with a selective integration scheme (selective SNS‐PFEM), and (3) SNS‐PFEM with a cubic bubble function and selective integration scheme (selective bSNS‐PFEM)—were gradually developed for comparison. The performance of these three approaches was first successively examined using two examples with elastic materials, that is, an infinite plate with a circular hole and Cook's membrane. The comparisons show that the cubic bubble function and selective integration scheme are both necessary as a locking‐free approach for modeling nearly incompressible materials, and the proposed selective bSNS‐PFEM performs best among the three variants in terms of accuracy and convergence. Two examples of slope stability analysis and footing penetration on elastoplastic materials were then conducted by SNS‐PFEM and the proposed selective bSNS‐PFEM. The results indicate that the proposed selective bSNS‐PFEM is stable and accurate, even when accompanied by significant deformation. All obtained results indicate that the locking‐free selective bSNS‐PFEM is a powerful approach for modeling nearly incompressible materials with both material and geometric nonlinearity.

A novel coupled NS‐PFEM with stable nodal integration and polynomial pressure projection for geotechnical problems

AbstractThe node‐based smoothed particle finite element method (NS‐PFEM) offers high computational efficiency but is numerically unstable due to possible spurious low‐energy mode in direct nodal integration (NI). Moreover, the NS‐PFEM has not been applied to hydromechanical coupled analysis. This study proposes an implicit stabilised T3 element‐based NS‐PFEM (stabilised node‐based smoothed particle finite element method [SNS‐PFEM]) for solving fully hydromechanical coupled geotechnical problems that (1) adopts the stable NI based on multiple stress points over the smooth domain to resolve the NI instability of NS‐PFEM, (2) implements the polynomial pressure projection (PPP) technique in the NI framework to cure possible spurious pore pressure oscillation in the undrained or incompressible limit and (3) expresses the NI for assembling coefficient matrices and calculating internal force in SNS‐PFEM with PPP as closed analytical expressions, guaranteeing computational accuracy and efficiency. Four classical benchmark tests (1D Terzaghi's consolidation, Mandel's problem, 2D strip footing consolidation and foundation on a vertical cut) are simulated and compared with analytical solutions or results from other numerical methods to validate the correctness and efficiency of the proposed approach. Finally, penetration of strip footing into soft soil is investigated, showing the outstanding performance the proposed approach can offer for large deformation problems. All results demonstrate that the proposed SNS‐PFEM with PPP is capable of tracking hydromechanical coupled geotechnical problems under small and large deformation with different drainage capacities.

Dynamic analysis of large deformation problems in saturated porous media by smoothed particle finite element method

This study presents a new formulation of smoothed particle finite element method (SPFEM) for dynamic problems in two-phase saturated porous media. A node integration method is used for both solid and water phases, which helps stabilize the coupled formulation when low-order triangle elements are used. Two different time integration schemes (i.e. explicit velocity Verlet and semi-implicit fractional step methods) are used to solve the coupled SPFEM formulations. In particular, a novel weakly-compressible fractional step algorithm is proposed, the key virtue of which is its capability to simulate high-frequency dynamic wave propagation in the porous media. The proposed coupled SPFEM is validated by comparing the numerical results with the analytical solution of two poroelastic examples, focusing on 1D wave propagation and 1D consolidation problems. Three more numerical tests are presented to further validate the proposed coupled SPFEM in simulating 2D wave propagation, self-weights slumping block and seepage flow induced progressive failure of embankment, and a very good agreement with the numerical results reported in the literature is obtained.

Two-phase PFEM with stable nodal integration for large deformation hydromechanical coupled geotechnical problems

The node-based strain smoothed particle finite element method (NS-PFEM) has a high level of computational efficiency. However, it has not yet been extended to hydromechanical coupling. This study presents the development of a two-phase one-point stable NS-PFEM (SNS-PFEM) for solving large deformation hydromechanical coupled geotechnical problems. First, the soil–water coupling is achieved by the v–w formulation based on Biot’s mixture theory, which automatically ensures the consistency between water pressure and stress that the former should be interpolated one order lower than that of velocity. Secondly, the node-based strain smoothing method with linear strain fields over the smoothed domain on 3-node triangular elements is adopted and implemented to obtain a stable nodal integration. Thirdly, a novel efficient water pressure stabilisation scheme based on an inverse distance interpolation algorithm is implemented to cure spurious spatial water pressure oscillations. The performance of the hydromechanical SNS-PFEM is primarily validated by both static and dynamic 1D consolidation and 2D wave propagation, and then applied to analyse the progressive landslide and the Selborne cutting slope involving large deformation. All results demonstrate that the proposed method is powerful and easily extensible for analysing large deformation hydromechanical problems in geotechnical engineering.

A mixed u\u2013p edge-based smoothed particle finite element formulation for viscous flow simulations

A mixed u–p edge-based smoothed particle finite element formulation is proposed for computational simulations of viscous flow. In order to improve the accuracy of the standard particle finite element method, edge-based and face-based smoothing operations on the displacement gradient are proposed for 2D and 3D analyses, respectively. Consequently, spatial integration involving the smoothing operator is performed on smoothing domains. The constitutive model is based on an elasto-viscoplastic formulation allowing for simulations of viscous fluid or fluid-like solid materials. The viscous response is modeled using an overstress function. The performance of the proposed edge-based smoothed particle finite element method (ES-PFEM) is demonstrated by several numerical benchmark studies, showing an excellent agreement with analytical and reference solutions and an improved accuracy and computational efficiency in comparison with results from the standard PFEM model. Finally, a numerical application of the ES-PFEM to the computational simulation of the extrusion process during 3D-concrete-printing is discussed.

NSPFEM2D: A lightweight 2D node-based smoothed particle finite element method code for modeling large deformation

The study presents a lightweight code that implements 2D node-based smoothed particle finite element method (NSPFEM) to fulfill the requirement of modeling large deformation in a variety of research fields and engineering applications. NSPFEM discretizes the problem domain with a set of Lagrangian nodes and employs Delaunay triangulation to build/re-build mesh on the nodes at each time step. It is therefore able to tackle large deformation without suffering from mesh entanglement. The method also adopts the nodal integration technique to avoid interpolation of material variables after remeshing which can eliminate interpolation-induced errors. The code has been built using hybrid C++ and Python programming, and is thus efficient and versatile to couple with other well-developed codes, such as MFront for the incorporation of various constitutive models and YADE, a discrete element method code, for multiscale simulations. The implementation of NSPFEM has first been validated using two examples, i.e., axial vibration of an elastic cantilever beam and a rolling cylinder down incline. Next, the capability and flexibility of the code has been demonstrated by three examples, namely retrogressive slope failure, pullout of a strip plate anchor, and multiscale modeling of trapdoor.

Interpretation of cone penetration test in clay with smoothed particle finite element method

Interpretation of cone penetration test in clay with smoothed particle finite element method

Large deformation failure analysis of slopes using the smoothed particle finite element method

Slope instability and landslides can be catastrophic events leading to loss of lives and properties. To prevent and assess the risks of slope failures, it is often desired that the dynamic process of the slope failure can be simulated, which is difficult with the classic Finite Element Method (FEM). In this study, the smoothed particle finite element method is developed based on the popular and efficient FEM. A numerical example with a slope model is employed to demonstrate the capacity of the proposed approach. An elastoplastic material model based on the Mohr–Coulomb yield criterion is used. The run out distance and failure mass is recorded which paves a way of being able to better quantify slope failure consequence and risk.

A coupled SPFEM/DEM approach for multiscale modeling of large‐deformation geomechanical problems

AbstractComputational modeling in geotechnical engineering frequently needs sophisticated constitutive models to describe prismatic behavior of geomaterials subjected to complex loading conditions, and meanwhile faces challenges to tackle large deformation in many geotechnical problems. The study presents a multiscale approach to address both challenges based on a hierarchical coupling of the smoothed particle finite element method (SPFEM) and the discrete element method (DEM) (coined SPFEM/DEM). In the approach, SPFEM serves as a solver for the global boundary value problem, in which the material constitutive responses are derived from the DEM solution of representative volume elements (RVEs) attached to the SPFEM nodes to avoid phenomenological constitutive assumptions. The approach is capable of modeling large deformation because of use of SPFEM, which discretizes the domain with a set of Lagrangian nodes and employs Delaunay triangulation for efficient remeshing on the nodes. In addition, as the RVEs are associated with the nodes due to the nodal integration technique in SPFEM, the interpolation of RVEs from the old mesh to the new one is bypassed, which is otherwise infeasible. The smoothing operation in nodal integration further offers a remedy for regularizing mesh dependency in simulation of strain localization problems. Two examples, namely, general failure of a footing and flow of an unstable slope, are used to demonstrate the potential of the proposed method in solving large deformation and providing reliable predictions on collapse and failure of geotechnical problems.