Material Point Method

The Material Point Method (MPM) is a version of the Particle-In-Cell (PIC) which has presented substantial advantage over the pure Lagrangian or Eulerian method in numerical simulations of problems involving large deformations. Using MPM helps to avoid mesh distortion and tangling problems related to Lagrangian methods and moreover the advection errors associated with Eulerian methods are avoided. MPM is also suits problems with complicated geometry and multiple materials in contact especially when the contact between deformable bodies is of interest. The variety of contact algorithms has been proposed to improve the accuracy of contact modelling by using MPM. Material point method has been applied to different problems such as granular material problems, fracture problems, fluid and fluid-structure interaction problems, geotechnical problems, and virtual simulation.
To model contact procedures like needle insertion and cutting brain, more accurate and reliable contact algorithms are required. Current contact algorithms are incapable of providing us with high fidelity results. One of the reasons could be lack of a precise normal vector definition on the contact area. More truthful results are attainable when either number of material points is increased or finer computational mesh is implemented. Both approaches are computationally expensive. Therefore, local enrichment/refinement is the solution and is not so computationally expensive. MPM compare to FEM is less accurate when it is dealing with small deformation. In order to settle this issue MPM and FEM is coupled.

MPM-1
Figure 1: Material point methodology, a: Penetration of a rigid rod into hyper-elastic body. b: background grid, material points and nodes. c: Snapshot of stress contour.

In MPM, the bodies of interest are represented by a group of Lagrangian particles, i.e. the material points, whereas the domain is discretised into an Eulerian computational grid, Figure 1. The material points are permitted to travel through the computational grid. The state variables, such as displacements, velocities, stresses and material properties are registered on the material points, although the computational grid is only utilised for solving governing equations and save no history of information when an explicit time integration scheme is applied. First the state variables are mapped from the material points to the nodes on the computational grid. Then the motion equations are solved on the computational grid to obtain the new nodal velocities. The data are mapped back on material point and then, the stress is computed through the constitutive law e.g. Hyper-elastic model. Lastly, the material points are relocated to its new position via updated velocities. Figure 1-c displays the snapshot of stress contour for the penetration of a rigid rod into a hyper-elastic body.
In order to use MPM with application on biomedical/biomechanical following objects are necessary to accomplish:
1- Robust and precise contact algorithm
2- Robust local enrichment and refinement for material points and computational grid, respectively to enhance the accuracy of MPM to handle small/large deformations
3- Optimum paralleling strategies to speed up the
4- Suitable shape functions for mapping data from material point method to background mesh and vice versa
To achieve the above objects a Hybrid FE-MP method is proposed, FE-MP method is furnished by novel contact algorithm based on truthful normal vector depends on contact conditions. Also a local enrichment/refinement strategy is utilized for Material point/computational to improve precision of the simulation. Coupling MPM with FEM is practicable due to existence of background mesh. Data from FEM is transferred to the material points through the nodes of the background grid and vice versa. Therefore a very smooth connection between FE and MP domains is utilised. The distorted elements located at FE domain are converted to MP particles based on a straightforward conversion scheme. The return journey from MP to FE, however, is a challenging problem and so this issue shall be investigated more through this project. The final goal would be converting FE domain to MP domain and vice versa. A Hybrid FE-MP method would be a reliable alternative for Arbitrary Lagrangian Eulerian method to be applied on Multiphase problems such as Fluid Structure Interaction (FSI). In order to parallelize Hybrid FE-MP method, PETSc library is to be used to have an effectual paralleling strategy.