This paper investigates the implementation of a nonlocal regularisation of the material point method to mitigate mesh-dependency issues for the simulation of large deformation problems in brittle soils. The adopted constitutive description corresponds to a simple elastoplastic model with nonlinear strain softening. A number of benchmark simulations, assuming static and dynamic conditions, were performed to show the importance of regularisation, as well as to assess the performance and robustness of the implemented nonlocal approach. The relevance of addressing stress oscillation issues, due to material points crossing element boundaries, is also demonstrated. The obtained results provide relevant insights into brittle materials undergoing large deformations within the MPM framework.

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Natural soil deposits may possess a highly anisotropic nature. The fabric anisotropy of soils which is induced during the soil formation process can lead to severe variation in field scale responses. Although the influence of fabric on the response of sands is well known and several advanced constitutive models have been developed to account for it, most of the studies incorporating anisotropy have focused on element test simulations while practical boundary value problem simulations are usually omitted. In this paper, the undrained response and liquefaction resistance of anisotropic sand deposits with different inherent fabric anisotropies are numerically investigated through element test simulations and one-dimensional nonlinear effective stress site response analyses. A novel semi-micromechanical constitutive model accounting for the effect of fabric anisotropy on sand liquefaction has been incorporated into a fully coupled dynamic in-house code employing the u-p formulation. The proposed numerical framework shows that, in both element test simulations and site response analyses, the fabric effects stemming from both the inherent and induced anisotropies can significantly influence the liquefaction resistance of sands.@en

Free-field site response analysis is a standard technique used to predict soil deposit dynamic response and liquefaction susceptibility. Such analyses are typically carried out by implementing periodic boundaries to guarantee the same speed of the dynamic waves travelling across them. However, when using random fields to consider the impact of soil spatial variability there is the possibility of an inconsistency with periodic boundaries. This is due to the generation of non-identical properties at the lateral boundaries on using traditional random fields. To overcome this inconsistency, this paper proposes periodic random fields to model spatial variability by matching the periodicity at the boundaries. To investigate the significance of using the proposed approach, a heterogeneous soil deposit subjected to earthquake loading is analysed using the random finite element method. The results show that, for certain values of the horizontal scale of fluctuation, ensuring consistency at the lateral boundaries could result in less conservative predictions of the extent of the liquefied areas. @en

The material point method (MPM) is gaining an increasing amount of attention due to its capacity to solve geotechnical problems involving large deformations. Large deformations in geotechnics usually involve the failure process and therefore dynamic analyses are often carried out. However, simulating the (infinite) continuous domain using typical Dirichlet (fixed) boundary conditions induces spurious reflections, causing (1) unrealistic stress increments at the domain boundary and (2) the appearance of multiple unnatural stress waves in the domain. Aiming to eliminate this numerical artifact in MPM, two solutions for absorbing boundary conditions found in FEM are implemented and investigated; these are (1) a viscous boundary condition and (2) a viscoelastic boundary condition. The use of such dynamic boundary conditions in MPM is scarce and no validation of them has yet been presented in the literature. In this paper, these absorbing conditions are implemented alongside recent mapping and integration techniques, improving numerical stability and accuracy.

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Soil liquefaction is investigated considering a saturated soil deposit and by implementing standard techniques of random field theory to distribute initial void ratio values and assess liquefaction risk. The soil domain is represented in a 2-dimensional (2D) random finite element model for the dynamic analysis of coupled behavior. Multiple Monte Carlo realizations are subjected to a base acceleration, while cyclic and small strain soil behaviours are achieved through a hypoplastic constitutive model. This investigation demonstrates that 2D stochastic simulations converge to 2D deterministic simulations when small standard deviations and/or small scales of fluctuation are used. However, large standard deviations combined with relatively large scales of fluctuation may cause significant uncertainty in the response of the soil deposit. Finally, common techniques employed to assess soil liquefaction are evaluated based on the results of the deterministic and random field analyses.

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1D soil column techniques are widely used to evaluate the potential of liquefaction in a system of soil layers. This approach generally leads to large inaccuracies since (1) soil layers are hardly homogeneous and perfectly horizontal and (2) horizontal effects are neglected. To demonstrate the limitation of 1D strategies and the need for 2D simulations, a series of benchmark problems are proposed and studied considering a fully coupled RFEM framework with small strain effects to account for cyclic behavior. First, a 1D simulation of a homogeneous material is tested against similar 1D problems including the spatial variation of soil properties (in this case void ratio). Then, a 2D domain is analyzed using the void ratio distribution obtained from combining the 1D columns. This investigation demonstrates that, by combining the effects of the horizontal direction and the spatial distribution of the soil properties, liquefaction triggering, spatial spreading and propagation extent may change significantly.@en

Mesh based methods such as the finite element method (FEM) are the most usually used techniques for analysing soil-structure interaction problems in geotechnical engineering. Nevertheless, standard FEM is unable to simulate large deformations and contact, hindering the realistic simulation of rotational, sliding, pull-out and overturning behaviours. Contemporary ‘particle’ methods, such as the material point method (MPM), do not use a mesh to discretise the material, allowing large deformations to be simulated. In this paper, a recently developed technique to simulate contact using implicit MPM is tested by simulating soil-structure interaction problems and a landslide. First, the behaviour of a retaining structure is studied during the impact of a mass of soil for different foundation conditions. Then, a landslide triggered by construction procedures is analysed. This new formulation allows the development of deep and shallow complex failure mechanisms (a combination of passive and active soil failures) and therefore the means to assess the consequences of a slope failure.

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An implicit contact algorithm for the material point method (MPM) has been developed to simulate contact. This allows recently developed implicit MPM codes to simulate large-scale deformations and interaction with external bodies. The performance of the method has been investigated and compared to an existing explicit method using benchmark and geotechnical examples. In particular, the proposed formulation has been shown to conserve energy in a similar way to the explicit formulation and reach similar results. The method typically converges to the analytical solution when an adequate time step and mesh size are used, with the time step generally being around ten times larger than the explicit method, although during the contact phase this does not always result in faster computation due to the iterative solution procedure.

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The material point method (MPM) shows promise for the simulation of large deformations in history-dependent materials such as soils. However, in general, it suffers from oscillations and inaccuracies due to its use of numerical integration and stress recovery at non-ideal locations. The development of a hydro-mechanical model, which does not suffer from oscillations is presented, including a number of benchmarks which prove its accuracy, robustness and numerical convergence. In this study, particular attention has been paid to the formulation of two-phase coupled material point method and the mitigation of volumetric locking caused numerical instability when using low-order finite elements for (nearly) incompressible problems. The numerical results show that the generalized interpolation material point (GIMP) method with selective reduced integration (SRI), patch recovery and composite material point method (CMPM) (named as GC-SRI-patch) is able to capture key processes such as pore pressure build-up and consolidation.@en

The material point method (MPM) is a numerical technique which has been demonstrated to be suitable for simulating numerous mechanical problems, particularly large deformation problems, while conserving mass,momentum and energy. MPM discretises material into points and solves the governing equations on a background mesh which discretises the domain space. The points are able to move through the mesh during the simulation. MPM is an improvement over other well-established numerical techniques, such as the finite element method (FEM), as it is able to simulate large deformations and therefore can simulate mechanical problems from initiation to the final outcome. It has the potential to become the preferred numerical tool to analyse many engineering problems. Nonetheless, it has been demonstrated throughout this thesis that the performance of MPM has often been far from the levels of accuracy desired in order to be considered a reliable technique for providing quantitative analyses for engineering problems. In this thesis, the implicit solution version of MPM has been taken as the starting point to investigate and solve its current main drawbacks, i.e. (i) the lack of accuracy when computing stresses (stress oscillations), and (ii) interaction between bodies, e.g. soil and structures. The stress oscillation problem is well-known in the MPM community, and is attributed mostly to material points crossing background cell boundaries, termed the cell-crossing problem. It has been shown in this thesis that cell-crossing is indeed one of the primary sources of oscillation. However, there are also other aspects contributing to the observed inaccuracies. In the literature, cell-crossing has been addressed by creating a particle domain, e.g. in the generalised interpolated material point (GIMP) method. It has been shown in this thesis that major problems also include (i) the use of linear shape function (SF) gradients to calculate (material point) strains and (ii) non-Gauss numerical quadrature to integrate material stiffness. The integration is made worse when using GIMP. In order to reduce the inaccuracies caused by integration a double mapping (DM) technique has been developed, which reduces the errors when integrating nodal stiffnesses. This is shown to also work well with GIMP (DM-G method). Additionally,DM has been combined with a Lagrangian interpolation technique, which uses a larger solution domain (through the combination of background cells to formpatches) to enhance the stresses computed at the material points (DM-C or DM-GC methods). The developed methods have been able to significantly improve the accuracy and stability of the simulated problems. This improvement will allow more robust use of more advanced constitutive models. The interaction of bodies is of benefit in large deformation simulations, although MPM can roughly simulate contact without special treatment. An MPM contact algorithm was initially proposed by other researchers for explicit time integration schemes, but no method was available for the implicit time integration scheme. An implicit contact scheme has been developed based on the original (explicit) contact formulation in order to calculate the change of nodal velocity during the Newton–Raphson iterative procedure. The results obtained with this contact methodology are shown to be as accurate as those computed using the explicit scheme, although generally with a larger time step. Additionally, it has been observed that, in most of the cases, implicit contact simulations are analysed faster than explicit simulations. However, the contact loads computed with this technique and the internal forces developed are inconsistent (i.e. not equal), reducing the energy conservation and remains an issue to be solved. An analysis of the problem is presented as a first step towards a solution. One challenge is that any method using consistent contact and internal forces is sensitive to stress oscillations, which can lead to highly unrealistic contact forces. Using the improvements developed in this thesis (i.e. DM-GC combined with the contact algorithm), soil-structure interaction problems and landslides have been successfully simulated. Incorporating the contact algorithm into the model has allowed the simulation of complex failure mechanism development during slope failure. The impact on neighbouring structures was realistic, and captured expected behaviours such as the sliding and rotation of the rigid elements. It has been demonstrated that (i) the accuracy in MPM has been improved via the combination of several (existing and novel) techniques, (ii) techniques developed for the explicit scheme (or other numerical methods) can be converted and introduced in implicit MPM, maintaining as much as possible the consistency of the formulation, and (iii) by improving diverse aspects of the formulation,more realistic simulations can be obtained. The work presented in this thesis makes several steps contributing to the improvement of MPM, which will lead towards it being used in engineering practice.@en

Stress inaccuracies (oscillations) are one of the main problems in the material point method (MPM), especially when advanced constitutive models are used. The origins of such oscillations are a combination of poor force and stiffness integration, stress recovery inaccuracies, and cell crossing problems. These are caused mainly by the use of shape function gradients and the use of material points for integration in MPM.The most common techniques developed to reduce stress oscillations consider adapting the shape function gradients so that they are continuous at the nodes. These techniques improve MPM, but problems remain, particularly in two and three dimensional cases. In this paper, the stress inaccuracies are investigated in detail, with particular reference to an implicit time integration scheme. Three modifications to MPM are implemented, and together these are able to remove almost all of the observed oscillations.

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Stress oscillations in the material point method (MPM) are one of the major reasons unrealistic results are obtained. In this paper an investigation of the stress oscillations occurring when using one- and two-phase approaches is performed. Specifically, an axisymmetric benchmark and a two-dimensional plane strain problem are used to demonstrate and investigate the oscillations. Furthermore, a partially reduced integration combined with the GIMP technique (GIMP-R) is implemented to reduce large oscillations. @en

In geotechnical engineering, proper design of retaining structures is of great importance, since failure of these structures can lead to catastrophic consequences. Nowadays, the finite element method is seen as a reliable numerical technique to analyze soil behaviour and is widely used to assess the interaction be-tween soil and rigid structures. However, a disadvantage of this method is the difficulty of simulating contact between separate bodies. Because of this, the event of a slope failing and colliding with a rigid body cannot be analyzed, so that the additional forces acting against the rigid body caused by the motion of the ground are neglected. With the recent development of the material point method (MPM), this limitation has been over-come and problems involving large deformation and multiple bodies in contact can be analyzed. In this paper, the effect of a landslide colliding with a rigid wall has been studied, and multiple initial conditions have been considered in order to identify the critical case.@en

Stress oscillations are a common phenomenon in the material point method (MPM), since this numerical method typically uses regular finite element (FE) shape functions to map variables from surrounding nodes to material points and vice versa, independently of the locations of the material points within an element. In integration this leads to a quadrature rule error and, in strain and stress calculations, derivatives of typical FE shape functions are discontinuous across element boundaries and do not give accurate results away from Gauss point locations within elements. In geotechnical analysis, the constitutive behaviour is generally stress-dependent, and therefore stress oscillations can cause severe inaccuracies in the calculated mechanical behaviour, including the development of wrong elasto-plastic quantities. Several attempts to improve stress recovery and reduce quadrature errors have been developed, but seldom has a full comparison between methods been made. In this paper, benchmark small scale and slope stability problems have been examined in order to compare the relative performance of the classic material point method (MPM), the generalized interpolation
material point (GIMP) method and the new compound material point method (CMPM)@en

Stress oscillations and inaccuracies are commonly reported in the material point method (MPM). This paper investigates the causes and presents a method to reduce them. The oscillations are shown to result from, at least in part, two distinctly different causes, both originating from the shape function approximations used. The first is due to the components of the stiffness matrix (or other matrices) being derived by performing numerical integration using the material points, and the second is due to calculating the strain from the nodal displacements of the elements, interpolated to the material points, via the shape function spatial derivatives. In this paper, an improvement for the recovery of results from the nodes to the material points is presented, where an increase of the Cn continuity along with an interpolation involving nodes from neighbouring elements is applied.@en