Comparison of a Material Point Method and a Galerkin Meshfree Method for the Simulation of Cohesive-Frictional Materials

. 2017 Sep 30;10(10):1150. doi: 10.3390/ma10101150

Algorithm 1 MPM algorithm.

(we will use

{(•)}^{n} = (•) (t_{n})

)

Material DATA: E,

ν

ρ

Initial data on material points:

m_{p}

x_{p}^{n}

Δ t

u_{p}^{n}, v_{p}^{n}, a_{p}^{n}, F_{p}^{n} = \sum_{I} \frac{\partial N_{I}}{\partial x_{I}^{0}} \cdot x_{I}^{n}

Δ F_{p} = \sum_{I} \frac{\partial N_{I}}{\partial x_{I}^{n}} \cdot x_{I}^{n + 1}

Initial data on nodes: NONE - everything is discarded in the initialization phase

OUTPUT of calculations:

Δ u_{I}^{n + 1}, σ_{p}^{n + 1}

INITIALIZATION PHASE
- Clear nodal info and recover undeformed grid configuration
- Calculation of initial nodal conditions.
  - (a)
    for p = 1: $N_{p}$
    *
    Calculation of nodal data
    
    ·
    $q_{I}^{n} = \sum_{p} N_{I} m_{p} v_{p}^{n}$
    
    ·
    $f_{I}^{n} = \sum_{p} N_{I} m_{p} a_{p}^{n}$
    
    ·
    $m_{I}^{n} = \sum_{p} N_{I} m_{p}$
  - (b)
    for I = 1: $N_{I}$
    
    *
    ${\tilde{v}}_{I}^{n} = \frac{q_{I}^{n}}{m_{I}^{n}}$
    
    *
    ${\tilde{a}}_{I}^{n} = \frac{f_{I}^{n}}{m_{I}^{n}}$
- Newmark method: PREDICTOR. Evaluation of $^{i t + 1} Δ u_{I}^{n + 1}$ , $^{i t + 1} v_{I}^{n + 1}$ and $^{i t + 1} a_{I}^{n + 1}$ using Equations (42)–(44)
UL-FEM PHASE
- for p = 1: $N_{p}$
  - (a)
    Evaluation of local residual ( $r h s$ ) (Equation (10))
  - (b)
    Evaluation of local Jacobian matrix of residual ( $l h s$ ) (Equation (25))
  - (c)
    Assemble rhs and lhs to the global vector $R H S$ and global matrix $L H S$ (Equations (30) and (37))
- Solving system $(Δ u_{I}^{n + 1})$
- Newmark method: CORRECTOR (Equations (43)–(45))
- Check convergence
  - (a)
    NOT converged: go to Step 2
  - (b)
    Converged: go to Step 3
CONVECTIVE PHASE
- Update the kinematics on the material points by means of an interpolation of nodal information (Equations (46)–(49))
- Save the stress $σ_{p}^{n + 1}$ , strain $ϵ_{p}^{n + 1}$ and total deformation gradient $F_{p}^{n + 1}$ on material points (the latter by $F_{p}^{n + 1} = Δ F_{p} \cdot F_{p}^{n}$ )