MPM Simulation

This project implements a 2D Material Point Method (MPM) simulation for elastic materials. It simulates the interaction between two circular disks using MPM techniques.

Overview

The Material Point Method is a numerical technique used to simulate the behavior of solids, fluids, and other materials. This implementation focuses on simulating elastic materials in a 2D space.

Key Components

1. Main Simulation (main.py):

   • Sets up the simulation parameters
   • Initializes the grid and particles
   • Runs the main simulation loop

2. MPM Solver (solver.py):

   • Implements the core MPM algorithm
   • Handles particle-to-grid and grid-to-particle transfers
   • Updates particle and grid properties each time step

3. Particles (particles.py):

   • Manages particle data and properties
   • Handles material assignment to particles

4. Grid (nodes.py):

   • Defines the background grid structure
   • Manages grid nodes and their properties

5. Material Models (material.py):

   • Implements material behavior (currently Linear Elastic)
   • Computes stress based on strain

6. Shape Functions (shape_function.py):

   • Defines shape functions for MPM interpolation
   • Includes functions for shape function gradients

7. Visualization (plotting.py, results_vis.py):

   • Provides functions to visualize the simulation results
   • Includes both static plotting and animation capabilities

8. Results Processing (results.py):

   • Handles saving simulation data to CSV files

9. Particle Shapes (particle_shapes.py):

   • Defines different particle shapes (e.g., Disk, Block)
   • Generates particles for each shape

Key Features

• Simulation of two colliding elastic disks
• Linear elastic material model
• Particle-based representation of materials
• Background grid for computation
• Visualization of particle positions, velocities, and stresses
• Energy tracking (kinetic, elastic, and total)
• Flexible particle shape generation (currently supports Disk and Block)

Usage

1. Set up the simulation parameters in main.py
2. Run main.py to start the simulation
3. Use results_vis.py to visualize the simulation results

Customization

• Adjust material properties in main.py
• Modify grid size and resolution in main.py
• Implement new material models in material.py
• Add boundary conditions in solver.py
• Create new particle shapes in particle_shapes.py

Output

The simulation generates CSV files for each time step, storing particle data including:

• Position
• Velocity
• Stress
• Strain
• Volume
• Mass
• Density
• Material properties

Visualization

The results_vis.py script provides an animated visualization of:

• Particle positions
• Velocity magnitudes
• Von Mises stress
• Energy evolution over time

Notes

• The current implementation focuses on 2D simulations
• Only linear elastic materials are implemented, but the structure allows for easy addition of other material models
• The simulation uses an explicit time integration scheme
• Particle shapes can be easily extended by subclassing the ParticleShape class

Future Improvements

• Implement more advanced material models (e.g., plasticity, damage)
• Add support for 3D simulations
• Implement adaptive time-stepping
• Optimize performance for larger simulations
• Add more boundary condition options
• Implement additional particle shapes

Dependencies

• NumPy
• Matplotlib
• tqdm (for progress bars)

Running the Simulation

1. Ensure all dependencies are installed
2. Run python main.py to start the simulation
3. After the simulation completes, run python results_vis.py <output_directory> to visualize the results

Visualization Options

The results_vis.py script now accepts command-line arguments for specifying the data directory. Use it as follows: