EXAGOOP - A material point method (MPM) solver based on AMReX Framework

Introduction

The EXAGOOP solver suite is a Material Point Method (MPM) based multi-phase solver developed by the HPACF team at National Renewable Energy Laboratory, Colorado. The solver, which can be broadly defined as a particle-based method, can be used to solve gaseous, liquid, and solid phase continuum, physical problems.

Most of the fluid and solid dynamics solvers are developed based on the Eulerian framework of the governing equations. These methods solve the governing equations on a collection of structured or unstructured grid elements. However, when materials undergo large deformations, the grid elements also stretch and deform, leading to inaccurate and often unstable computations. On the other hand, MPMs, like many particle-based methods, is based on the Lagrangian framework of the governing equations. The continuum material under study is modeled as a collection of particles (or material points). Unlike the Eulerian solvers, the material's properties are stored on these material points. The Lagrangian form of the governing equations are solved using the material points and a reference background grid. This eliminates the problem posed by grid element deformations. Hence, MPM methods are well suited for all continuum mechanics problems in general and for those involving large material deformations, in particular.

The basic components and the terminologies used in EXAGOOP MPM solver is shown in the figure below. A cartesian Eulerian grid (no adaptive mesh refinement) is used along with material points simulated as spherical particles.

The various steps involved in one time integration stage in EXAGOOP is shown in the following figure and are described in the following 4 steps.

Steps:

0. In the initialization stage, material point mass, position, velocity and stresses are initialized.
1. The material point (subscript p) mass and momentum are mapped onto the grid node (subscript I) using grid shape functions $\phi$. Similarly, the forces on the material points (comprising external forces such as gravity and internal forces from stresses) are also mapped to grid nodes. Mathematically,

$$ m_I^t = \sum_p \Phi_I (x_p^t) m_p $$

$$ (m\mathbf{v})_I^t = \sum_p \Phi_I (x_p^t) (mv)_p $$

$$ \mathbf{f}_I^{ext,t} = \sum_p \Phi_I (x_p^t) m_p \mathbf{b}(x_p) $$

$$ \mathbf{f}_I^{int,t} = \sum_p V_p^t \mathbf{\sigma}_p^t \nabla \Phi_I (x_p^t) $$

where, $\sum_p$ denotes summation over all material points. $m_p$, $\mathbf{v}$, $V_p$, $\mathbf{b}$, $\sigma$ and $\Phi$ denote particle mass, particle velocity vector, particle volume, body force vector, stress and grid shape functions respectively.

2. The updated grid nodal velocity is calculated using explicit Euler time integration.

$$ \mathbf{v}_I^{t+\Delta t} = \mathbf{v}_I^{t} + \frac{\mathbf{f}_I^{ext,t}+\mathbf{f}_I^{int,t}}{m_I^t} \Delta t $$

3. Particle velocity is updated from new nodal velocity (at time $t+\Delta t$) using a blend of Particle in Cell (PIC) and Fluid Implicit Particle (FLIP) update methods. The blending factor used is $\alpha$.

$$ \mathbf{v}_p^{t+\Delta t}=\alpha\left(\mathbf{v}_p^t+\sum_I \Phi_I\left[\mathbf{v}_I^{t+\Delta t}-\mathbf{v}_I^t\right]\right)+\left(1-\alpha\right) \sum_I \Phi_I \mathbf{v}_I^{t+\Delta t} \nabla \mathbf{v}_p^{t+\Delta t}=\sum_I^{n g} \nabla \Phi_I \mathbf{v}_I^{t+\Delta t} $$

4. Particle positions are updated using Euler time integration.

$$ x_p^{{t+\Delta t}}=x_p^t+\Delta t \sum_I \phi_I\left(x_p^t\right) v_I^{{t+\Delta t}} $$

EXAGOOP features

• Based on AMReX framework. Single-level, block-structured, cartesian grid is used as the background grid. Material points are simulated using particle class in AMReX.
• MPI+GPU support using CUDA and HIP
• Linearly elastic and compressible fluid material models available
• Linear, Quadratic, and Cubic B-Spline grid shape functions available
• Explicit time integration used for time update
• Complex geometry simulated using embedded boundary method
• Rigid material points available for simulating stationary/moving rigid walls

Build Instructions

• Clone AMReX source code to a convenient location and point AMREX_HOME environment variable to this directory
• Clone EXAGOOP from GitHub to a convenient location and set MPM_HOME enviroment variable to this directory
• Go to BUILD directory. Modify the GNUmake file according to the problem to be simulated. Set COMP to GNU and USE_MPI to TRUE for MPI support. For GPU support, enable USE_CUDA variable.
• Run make to compile and generate the executable (One can test running the executable using the test cases in the tests folder).

Visualization Instructions

• The simulation output files are in the form of AMReX plotfiles.
• Paraview can be used to load and view the particle ( plt files) and nodal ( nplt files) solution files.