A stable node-based smoothed PFEM for solving geotechnical large deformation 2D problems

SummaryTo solve large deformation geotechnical problems, a novel strain‐smoothed particle finite element method (SPFEM) is proposed that incorporates a simple and effective edge‐based strain smoothing method within the framework of original PFEM. Compared with the original PFEM, the proposed novel SPFEM can solve the volumetric locking problem like previously developed node‐based smoothed PFEM when lower‐order triangular element is used. Compared with the node‐based smoothed PFEM known as “overly soft” or underestimation property, the proposed SPFEM offers super‐convergent and very accurate solutions due to the implementation of edge‐based strain smoothing method. To guarantee the computational stability, the proposed SPFEM uses an explicit time integration scheme and adopts an adaptive updating time step. Performance of the proposed SPFEM for geotechnical problems is first examined by four benchmark numerical examples: (a) bar vibrations, (b) large settlement of strip footing, (c) collapse of aluminium bars column, and (d) failure of a homogeneous soil slope. Finally, the progressive failure of slope of sensitive clay is simulated using the proposed SPFEM to show its outstanding performance in solving large deformation geotechnical problems. All results demonstrate that the novel SPFEM is a powerful and easily extensible numerical method for analysing large deformation problems in geotechnical engineering.

The 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.

In this paper, a meshfree method called adaptive CTM–RPIM is developed to model geotechnical problems with large deformation. The developed adaptive CTM–RPIM is a combination of the Cartesian transformation method (CTM), the radial point interpolation method (RPIM) and the alpha shape method. To reduce the requirement for meshes, the CTM is adopted to transform domain integrals into line integrals, and the RPIM is applied to construct interpolation functions. The alpha shape method, which is capable of capturing severe boundary evolution due to large deformations, is then introduced into the CTM–RPIM to form the adaptive CTM–RPIM. The accuracy of CTM–RPIM is first verified by considering a cantilever beam under small deformation, where the influence of key parameters on the simulation results is explored. Afterward, the ability of the adaptive CTM–RPIM to handle large deformation problems is demonstrated by simulating cantilever beams with large deformations for which analytical solutions are available. Finally, the ability of the proposed method to model the geotechnical large deformations is illustrated from both quasi-static and dynamic aspects, where a slope failure problem and a footing bearing capacity problem are modeled to evaluate the stability of geotechnical structures; and a 2-D soil collapse experiment using small aluminum bars is simulated to show the method capability in describing the soil flows. These benchmark examples demonstrate that the adaptive CTM–RPIM is a numerical method with broad application prospects for modeling large deformation problems in geotechnical engineering.

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

In this study, a simple PFEM approach for analyzing large deformation problems in geotechnical practice is implemented in the commercial FEM package Abaqus. The main feature of the proposed Abaqus-PFEM approach lies in its capability to absorb the advantages of the functionality available in Abaqus and integrate them into PFEM with a single Python script, which leads to a considerable reduction in coding work. By utilizing the built-in functions in Abaqus to fulfil the standard incremental FEM analysis, as well as the powerful mesh-to-mesh solution mapping technique, the proposed Abaqus-PFEM approach allows for the large deformation analysis automatically running with a single Python script and requires no intervention from the user. The accuracy of the proposed Abaqus-PFEM approach is firstly validated through a simple elastic cantilever beam bending problem. Then, the performance and robustness of the proposed Abaqus-PFEM approach are further examined by three illustrative numerical examples: penetration of rigid footing, penetration of T-bar and pipeline–soil interaction problem. The numerical results demonstrate that the proposed Abaqus-PFEM approach as a powerful and easily extensible numerical tool is capable of handling large deformation and soil–structure interaction problems in geotechnical engineering, and consequently, it offers an alternative way to tackle such problems.

T. Wichtmann, T. Triantafyllidis: Behaviour of granular soils under environmentally induced cyclic loads. - D. Muir Wood: Constitutive modelling. - C. di Prisco: Creep versus transient loading effects in geotechnical problems. - M. Pastor et al.: Mathematical models for transient, dynamic and cyclic problems in geotechnical engineering. - M. Pastor: Discretization techniques for transient, dynamics and cyclic problems in geotechnical engineering: first order hyperbolic partial diffential equations. - M. Pastor et l.: Discretization techniques for transient, dynamic and cyclic problems in geotechnical engineering: second order equation. - C. di Prisco: Cyclic mechanical response of rigid bodies interacting with sand strata. - D. Muir Wood: Macroelement modelling. - M. F. Randolph: Offshore design approaches and model tests for sub-failure cyclic loading of foundations. - M.F. Randolph: Cyclic interface shearing in sand and cemented solis and application to axial response of piles. - M. F. Randolph: Evaluation of the remoulded shear strength of offshore clays and application to pipline-soil and riser-soil interaction.

Trivariate analysis of soil ranking-correlated characteristics and its application to probabilistic stability assessments in geotechnical engineering problems

The integration of numerical modeling and physical measurements through inverse analysis in geotechnical engineering

Numerical modeling of geotechnical problems frequently requires the consideration of large deformation soil responses. The Material Point Method (MPM) has gained increasing popularity recently over many other methods such as Finite Element Method (FEM) in continuum modeling of large deformation problems in geomechanics. This study presents a new enrichment of MPM with multi-scale predictive capabilities. We propose a computational multiscale scheme based on coupled MPM and DEM following a similar concept of the FEM-DEM coupling scheme [2, 3]. The MPM is employed to treat a typical boundary value problem in geomechanics that may experience large deformations, and the DEM is used to derive the nonlinear material response required by MPM for each of its material points. The proposed coupling framework helps avoid phenomenological constitutive assumptions in typical MPM, while inherits its advantageous features in tackling large deformation problems over FEM (e.g., no need for re-meshing to avoid highly distorted mesh in FEM). Importantly, it provides direct micro- macro linking for us to understand complicated behavioral changes of granular media over all deformation levels, from initial elastic stage en route to large deformation regime before failure. Several demonstrative examples are shown to highlight the advantages of the new MPM-DEM framework, including the collapse of a soil column, biaxial shear tests and different failure modes observed in a footing problem in geotechnical problems.

This paper presents a novel semi-implicit two-phase double-point Material Point Method (MPM) for modelling large deformation in geotechnical engineering problems. To overcome the computational limitations of explicit methods, we develop a semi-implicit approach that eliminates time step dependency in soil–water coupled problems. Unlike existing single-point methods that use one set of material points to represent soil–water mixtures, our approach employs two distinct sets of material points to model soil and water phases separately. To address MPM’s inherent stress oscillations, we introduce a stabilisation technique based on the modified F-bar method. Through validation against Terzaghi’s one-dimensional consolidation theory, one-dimensional large deformation consolidation, and large deformation slope stability studies, our method demonstrates superior performance. Further testing with the hyperelastic Nor-Sand constitutive model in landslide simulations reveals that the double-point approach produces significantly more reliable results than single-point methods, particularly for dilatant soils. Notably, while implementing two sets of material points, our method incurs less than 10% increase in computational cost while achieving markedly improved accuracy. These findings establish the double-point MPM as a robust and efficient approach for analysing large deformation geotechnical problems under fully saturated conditions.

In this paper we introduce various forms of strain smoothing for stabilization and regularization of two types of instability: (1) numerical instability resulting from nodal domain integration of weak form, and (2) material instability due to material strain softening and localization behavior. For numerical spatial instability, we show that the conforming strain smoothing in stabilized conforming nodal integration only suppresses zero energy modes resulting from nodal domain integration. When the spurious nonzero energy modes are excited, additional stabilization is proposed. For problems involving strain softening and localization, regularization of the ill-posed problem is needed. We show that the gradient type regularization method for strain softening and localization can be formulated implicitly by introducing a gradient reproducing kernel strain smoothing. It is also demonstrated that the gradient reproducing kernel strain smoothing also provides a stabilization to the nodally integrated stiffness matrix. For application to modeling of fragment penetration processes, a nonconforming strain smoothing as a simplification of conforming strain smoothing is also introduced.

Domain integration by Gauss quadrature in the Galerkin mesh-free methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the mesh-free Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. No numerical control parameter is involved in the proposed strain smoothing stabilization. The numerical results show that the accuracy and convergent rates in the mesh-free method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in mesh-free discretization. For this reason the proposed method provides even better accuracy than Gauss integration for Galerkin mesh-free method as presented in several numerical examples. Copyright © 2001 John Wiley & Sons, Ltd.

On the basis of the moving least-squares (MLS) approximation, the complex variable element-free Galerkin (CVEFG) method for two-dimensional elastic large deformation problems, such as large displacement and large rotation, is presented in this paper. The advantage of the CVMLS approximation is that the trial function of a two-dimensional problem is formed with one-dimensional basis function. Then the CVMLS approximation is applied to solve two-dimensional large deformation problems. The Galerkin weak form is employed to obtain the discretized system equation, the penalty method is used to apply the essential boundary conditions. And then the complex variable element-free Galerkin (CVEFG) method for two-dimensional large deformation problems based on the total lagrange formulation is formed, the corresponding formulae are obtained, and the Newton-Raphson iteration is used in the numerical implementation. Three numerical examples are given to show that this method in this paper is effective for elastic large deformation problems.

Settlement of embankments on soft soils is a significant problem in geotechnical engineering to maintain pavements, buildings, and other facilities on them. The problem that often arises is not being able to predict the magnitude of consolidation compression accurately, this occurs due to uncertainty in field conditions, laboratory testing, and data interpretation, as well as assumptions made in the development of 1-D consolidation theory. Based on the causes of inaccurate settlements predictions, it is necessary to carry out research on a better method for predicting embankment’s settlement on soft soil. To obtain the correct method, a study was conducted by comparing and examining the consolidation settlement of embankments built on soft soil using theoretical calculations and field measurement results. The process involved monitoring the settlements using 25 plate measurement data in embankment preloading for housing and building construction over the very soft clay. The results showed the compression parameter, especially Cc, is very influential on the compression ratio, and the Cc value based on the empirical formula of Bowles (1989) turned out to be the most suitable for the actual compression results in the field with a compression ratio between 0.6-1.1 and confidence of level 90%.

Large deformation problems arising from deep excavation in silt strata: A case study in Shenzhen, China