# Formulation and implementation of two-phase Material Point Method

**Introduction**

The Finite Element Method (FEM) is a standard tool in engineering science especially in the field of geotechnical engineering. This method has been widely used to investigate different applications in this field. FEM has shortcomings when dealing with large deformations of the material (soil) and mesh distortions are common in this area (Figure 1). To be able to study applications with large deformations or displacements like the failure of the slopes or pile driving, simulation methods are needed which can overcome these shortcomings. Groups of methods called the mesh free methods or the mesh based particle methods have been developed by different researchers to overcome the mesh tangling problem of FEM. Material Point Method (MPM) is a mesh based particle method which is developed in the last decades and is the application of Particle in Cell (PIC) method from fluid to solid mechanics. MPM is a powerful tool to simulate large deformation problems and excludes the drawbacks appearing in the conventional FEM.

Figure 1. Distorted FEM mesh under large deformation [Beuth 2012]

**Material Point Method (MPM)**

MPM discretizes the continuum using the material points or particles and discretizes the space using an Eulerian background fixed mesh on which the equations of motion are solved. This mesh should cover the whole space where the material may go during the simulation process. Material points are the integration points which can move during the simulation and using this property MPM is able to analyze large deformations or displacements of the material. Particles carry all the physical information (e.g. stresses, strain, etc.) during the simulation process and no permanent data are stored on the mesh.

Figure 2. Working procedure of MPM

Working procedure of MPM for one time step consists of three phases (Figure 2). First is the initialization phase where all the data are mapped from particles to the nodes of the background mesh using the nodal shape functions. Then is the Lagrangian phase in which the equations of motion are solved on the mesh. At the end is the convective phase. In this phase information are mapped back from the mesh to the material points and updates the data of the particles. Background mesh goes back to its original position. As an example, Figure 3 shows the total displacements of particles in a slope failure (left) and a strip footing failure (right) simulated using one phase MPM.

Figure 3. Simulation of failure of a footing using single phase MPM [Fatemizadeh 2014] (left) and simulation of bearing capacity of a strip footing using single phase MPM [Fatemizadeh, Hamad & Moormann 2015] (right)

**Two-phase MPM**

Considering the field of geotechnical engineering, ground water plays an important role on the mechanical behavior of soils. This effect should be taken into account in order to have a better understanding of different phenomena in this field. Two-phase formulation of FEM is fully described and validated by different researchers, but considering the large deformation of fully saturated soils, FEM is not a proper tool. The formulation of two-phase MPM considering the solid phase as well as the pore water and their interaction is needed to be investigated as a robust numerical method for studying different applications including large deformations in saturated soils (Figure 4). Examples of such applications are the hydraulic failure in excavations, erosion processes, etc.

Figure 4. Dry (left), fully saturated (middle) and partially saturated (right) soils

As the weak formulation of MPM is the same as FEM, the governing equations of FEM can be adopted for MPM. In order to describe the fully coupled behavior of saturated soils, the u-U formulation is preferred in which the displacement of solid phase and the water phase are primary variables.

In the context of MPM for fully saturated soils there are two implementations possible. First is using one kind of particle to represent the mixture of solid grains and water (Figure 5). Second is using one kind of particle to represent solid grains and the other to represent the pore water (Figure 6). Both implementations have their own advantages and disadvantages.

Figure 5. Representing two-phase formulation in MPM using one kind of particle

Figure 6. Representing two-phase formulation in MPM using two kinds of particles (right)

**Future work**

During this project, the two-phase formulation of MPM will be investigated and a 2D program will be developed based on the considered formulation. Demonstrating examples will be solved and different benchmark problems consisting soil-water interaction will be studied.

**References**

1] Beuth, L., Formulation and Application of a Quasi-Static Material Point Method, PhD thesis, Institute of Geotechnical Engineering, University of Stuttgart, 2012.

[2] Fatemizadeh, S., Implicit Implementation of Material Point Method for Simulation of Incompressible Problems in Geomechanics, 33. Baugrundtagung: Forum für junge Geotechnik-Ingenieure, Berlin, 2014, S. 45-51.

[3] Fatemizadeh, S., Hamad, F. & Moormann, C., Effects of the MPM discretisations on soil-structure problems, 2015.

[4] Jassim, I.K., Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems, PhD thesis, Institute of Geotechnical Engineering, University of Stuttgart, 2013.

[5] Sulsky, D., Zhou, S.J. & Schreyer, H.L. Application of a particle-in-cell method to solid mechanics, Computer Physics Communications, Vol. 87, 236-252, 1995.

contact: Farzad Fatemizadeh