Material Point Method (MPM)

Geophysical hazards usually involve multiphase flow of dense granular solids and water. Understanding the mechanics of granular flow is of particular importance in predicting the run-out behaviour of debris flows. The dynamics of a homogeneous granular flow involve three distinct scales: the microscopic scale, the meso-scale, and the macroscopic scale.

Conventionally, granular flows are modelled as a continuum because they exhibit many collective phenomena. ecent studies, however, suggest that a continuum law may be unable to capture
the effect of inhomogeneities at the grain scale level, such as orientation of force chains, which are micro-structural effects. Discrete element methods (DEM) are capable of simulating these micro-structural effects, however they are computationally expensive. In the present study, a multi-scale approach is adopted, using both DEM and continuum techniques, to better understand the rheology of granular flows and the limitations of continuum models.

The collapse of a granular column on a horizontal surface is a simple case of granular flow; however, a proper model that describes the flow dynamics is still lacking. In the present study, the generalised interpolation material point method
(GIMPM), a hybrid Eulerian -- Lagrangian approach, is implemented with the Mohr-Coloumb failure criterion to describe the continuum behaviour of granular flows. The granular column collapse is also simulated using DEM to understand the micro-mechanics of the flow.

The limitations of MPM in modelling the flow dynamics are
studied by inspecting the energy dissipation mechanisms. The lack of collisional dissipation in the Mohr-Coloumb model results in longer run-out distances for granular flows in dilute regimes (where the mean pressure is low). However, the model is able to capture the rheology of dense granular flows, such as the run-out evolution of slopes subjected to lateral excitation, where the inertial number I < 0.1.

The initiation and propagation of submarine flows depend mainly on the slope, density, and quantity of the material destabilised. Certain macroscopic models are able to capture simple mechanical behaviours, however the complex physical mechanisms that occur at the grain scale, such as hydrodynamic
instabilities and formation of clusters, have largely been ignored. In order to describe the mechanism of submarine granular flows, it is important to consider both the dynamics of the solid phase and the role of the ambient fluid. In the present study, a two-dimensional coupled Lattice Boltzmann LBM -- DEM technique is developed to understand the micro-scale
rheology of granular flows in fluid. Parametric analyses are performed to assess the influence of initial configuration, permeability, and slope of the inclined plane on the flow. The effect of hydrodynamic forces on the run-out evolution is analysed by comparing the energy dissipation and flow
evolution between dry and immersed conditions.

The application of a new Material Point Method (MPM) approach to model the proppant distribution in a reservoir where hydraulic fractures interact with natural fractures is presented and validated with an Eagle Ford well. The new MPM approach uses particles to represent the slurry and its effects on the hydraulic and natural fractures. The particles are injected in the hydraulic fractures and their action causes the hydraulic fractures to propagate and interact with the natural fractures thus providing new pathways for the proppant to move away from the wellbore when optimal natural fracture orientations are encountered. Elementary tests, conducted with the new MPM approach show that fractures oriented in certain directions close to the hydraulic fracture orientation facilitate the proppant placement while other fracture orientations that are perpendicular to the hydraulic fracture or close to perpendicular direction appear to cause screen out. When using the new MPM approach on multiple interacting natural fractures with different orientations the same conclusions observed with one single fractures still hold and only those close to the orientation of the hydraulic fractures promote the placement of the proppant. The application of the new technology to an Eagle Ford well shows that the simulated proppant travels the farthest in the frac stage that is known from other methods and measurements to be the one with the best half fracture length. Furthermore, the examination of the simulated proppant concentration curve versus time shows striking similarities with the actual slurry concentration measured at the same well. These encouraging results show that the macroscopic modeling of proppant distribution using the new MPM technology could help improve our understanding of the complex hydraulic fracturing process and the resulting proppant distribution.

The Material Point Method (MPM) uses two kinds of space iscretization. One is physical space, in which physical quantities are evaluated, and the other is calculation space, in which discretized domain movements are solved. In the course of the MPM process, physical quantities are mapped from material points to a numerical grid using interpolation functions. On the contrary, material points recover physical quantities from their motions, which are mapped from the numerical grid to material points using interpolation functions. In its original form, MPM treats each material point as a point. Later, an enhanced method called the Generalized Interpolation Material Point (GIMP) method was roposed in which the effect of the size of the controlling domain is considered. More recently, the Convected Particle Domain Interpolation (CPDI) method has been proposed, in which a material point is considered to be an arbitrarily shaped controlling domain. Thus MPM and its derivatives are gradually gaining improved and enhanced numerical characteristics.
The CPDI method, which is able to consider arbitrarily shaped domain integration, has been formulated as an interpolation function for a two-dimensional controlling domain using a direct integration technique. However, it is difficult to extend direct integration to three-dimensional problems because of a drastic increase in the number of terms, leading to computational complexity. In this paper, the authors employ a numerical integration technique for domain integration over the material points in the CPDI method. Called the Arbitrary Particle Domain Interpolation (APDI) method, it has been applied to threedimensional analysis. The APDI method allows use of the same numerical procedure even if material points have controlling domains of different shapes. Through a demonstrative analysis of two- and threedimensional numerical problems related to geomaterials, the applicability and effectiveness of the APDI method are clarified.

Modern numerical techniques utilised to model crack propagation tend to be optimized for tracking the evolution of a single crack. Real fracture processes are however complex, involving the initiation and propagation of opening (activated) cracks, while other may close (deactivate) and undergo frictional dissipations. Accounting for the correct loss of energy (through debonding and friction) is essential to achieving a realistic description of the fracture process. One common strategy has been to make small adaptations to traditional techniques to tackle multiple cracking, in effect relying on extensive complicated computational algorithms. A typical example is the use of cohesive models in combination with the eXtended finite Element method where cracks, sometimes intersecting, need to be defined explicitly. In this study the Material Point Method is used for the analysis of fracture propagation. Crack states, as internal variables, are stored within the material points and mapped as strong discontinuities to the elements during the Lagrangian phase of the solution. Consequently, material points carrying cracks of different sizes and orientations are allowed to cohabit within the same element, yielding a natural description of the fracture/fragmentation process. The three-point bending test is used to demonstrate the features of the new approach.

Numerical simulations, which include spacial descretization, difine the controlling domain of descrete elements, according to each characteristics of methods. Finite Element Method uses nodes and elements to define their controlling domains, whereas particle based methods use the interpolation functions. Parti-cles based methods usually updates its geometric position, while they continue to use the same interpola-tion function through the whole simulation process. In particle based methods, separations due to numerical algorithms often occur when the intensive streches act on the materials, not because materials reaches its criteria of separation but because the distance from neigbor particles exceeds the interpolation radius.
The same difficulty has been discussed over the Material Point Method. Although original MPM, which generates numerical noise when crossing numerical grids, does not consider a controlling domain, GIMP method, which considers the change of controlling domain in axial direction, has been proposed to overcome the problems. More recentry, CPDI method, which considers the whole change of controlling domain, has been proposed. In this paper the author focuses on CPDI method and newly propeses the three dimensional formulation of CPDI method and applies it to the large deformation problem of geo-materials.

"Avalanches, debris flows, and landslides are geophysical hazards, which involve rapid mass movement of granular solids, water and air as a single-phase system. The dynamics of a granular flow involve at least three distinct scales: the micro-scale, meso-scale, and the macro-scale. This study aims to  understand the ability of continuum models to capture the micro-mechanics of dry granular collapse. Material Point Method (MPM), a hybrid Lagrangian and Eulerian approach, with Mohr-Coulomb failure criterion is used to describe the continuum behaviour of granular column collapse, while the micromechanics is captured using Discrete Element Method (DEM) with tangential contact force model. The run-out profile predicted by the continuum simulations matches with DEM simulations for columns with small aspect ratios (‘h/r’ < 2), however MPM predicts larger run-out distances for columns with higher aspect ratios (‘h/r’ > 2). Energy evolution studies in DEM simulations reveal higher collisional dissipation in the initial free-fall regime for tall columns. The lack of a collisional energy dissipation mechanism in MPM simulations results in larger run-out distances. Micro-structural effects, such as shear band formations, were observed both in DEM and MPM simulations. A sliding flow regime is observed above the distinct passive zone at the core of the column. Velocity profiles obtained from both the scales are compared to understand the reason for a slow flow run-out mobilization in MPM simulations.

Contact between deformable bodies is a dif- cult problem in the analysis of engineering systems. A new approach to contact has been implemented us- ing the Material Point Method for solid mechanics, Bar- denhagen, Brackbill, and Sulsky (2000a). Here two im- provements to the algorithm are described. The rst is to include the normal traction in the contact logic to

A material point method (MPM)-based numerical model has been used to study the effect of asperity size and orientation relative to sliding direction on the ploughing behaviour of a rigid, ellipsoidal asperity. Based on the simulated ploughing behaviour, an analytical model has been extended to calculate the ploughing depths over the wear track and compute the forces acting over the contacting surface of an ellipsoidal asperity sliding through a rigid-plastic substrate. The analytical model results have been compared with the MPM model results. The MPM model results are also validated to be in good agreement with the friction forces and ploughing depths measured from the ploughing experiments on lubricated steel sheets with ellipsoidal indenters up to certain sizes and orientations.

We develop a novel constitutive modeling approach for the analysis of fracture propagation in quasi-brittle materials using the Material Point Method. The kinematics of constitutive models is enriched with an additional mode of localized deformation to take into account the strain discontinuity once cracking has occurred. The crack details therefore can be stored at material point level and there is no need to enrich the kinematics of finite elements to capture the localization caused by fracturing processes. This enhancement also removes the drawback of classical smeared crack approach in producing unphysical snapping back constitutive responses when the spatial resolution is not fine enough. All these facilitate the implementation of the new approach in the Material Point Method for analysis of large scale problems. Numerical examples of fracture propagation are used to demonstrate the effectiveness and potentials of the new approach.

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