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index 0000000..c45e04e
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diff --git a/README.md b/README.md
index 370ec15..00c38ad 100644
--- a/README.md
+++ b/README.md
@@ -1 +1,110 @@
-# py_mpm
\ No newline at end of file
+# 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
+
+## 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)
+
+## 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`
+
+## 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
+
+## 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
+
+## 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` to visualize the results
\ No newline at end of file
diff --git a/main.py b/main.py
new file mode 100644
index 0000000..a8d01ef
--- /dev/null
+++ b/main.py
@@ -0,0 +1,123 @@
+import os
+import csv
+import numpy as np
+from tqdm import tqdm
+from particle_shapes import Disk
+from nodes import Grid
+from particles import Particles
+from solver import MPMSolver
+from plotting import visualize_particles_and_grid
+from results import save_particles_to_csv
+# import projections
+
+# Setup grid parameters
+grid_size = (0.6, 1)  # Adjust as needed
+cell_size = 0.01  # Adjust based on your simulation scale
+
+# Initialize the grid
+grid = Grid(grid_size, cell_size)
+
+# Calculate the physical size of the grid
+physical_size = (grid_size[0] * cell_size, grid_size[1] * cell_size)
+
+print(f"Grid initialized with size {grid_size} and physical dimensions {physical_size}")
+print(f"Total number of nodes: {grid.total_nodes}")  # We now have 2500 nodes (50 * 50)
+
+# Nodes can be accessed using grid.nodes
+# For example, to access the node at position (i, j):
+# node = grid.get_node(i, j)
+
+# Initialize Material Points
+
+
+# Define disk parameters
+disk1_radius = 8*cell_size
+disk1_center = (0.2, 0.5)
+disk1_object_id = 1
+
+disk2_radius = 8*cell_size
+# Calculate the center of the second disk
+# It should be 2 cells from the edge of the first disk
+disk2_x = disk1_center[0] + disk1_radius + 3*cell_size + disk2_radius
+disk2_center = (disk2_x, disk1_center[1])
+disk2_object_id = 2
+
+# Create Disk objects
+disk1 = Disk(cell_size, disk1_radius, disk1_center, disk1_object_id)
+disk2 = Disk(cell_size, disk2_radius, disk2_center, disk2_object_id)
+
+# Generate particles for both disks
+particles1 = disk1.generate_particles()
+particles2 = disk2.generate_particles()
+combined_particles = particles1 + particles2
+
+# Define material properties
+material_properties = {
+    1: {  # Object ID 1
+        "type": "LinearElastic",
+        "density": 1000,  # kg/m^3
+        "youngs_modulus": 1e6,  # Pa
+        "poisson_ratio": 0.3
+    },
+    2: {  # Object ID 2
+        "type": "LinearElastic",
+        "density": 1000,  # kg/m^3
+        "youngs_modulus": 1e6,  # Pa
+        "poisson_ratio": 0.3
+    }
+}
+
+# Create Particles object
+particles = Particles(combined_particles, material_properties)
+
+# Set initial velocities for each object
+particles.set_object_velocity(object_id=1, velocity=[2.0, 0.0])  # Object 1 moving right
+particles.set_object_velocity(object_id=2, velocity=[-2.0, 0.0])  # Object 2 moving left
+
+print(f"Generated {len(particles1)} particles for disk 1")
+print(f"Generated {len(particles2)} particles for disk 2")
+print(f"Total particles: {particles.get_particle_count()}")
+
+# # Call the visualization function
+visualize_particles_and_grid(particles, grid, 0)
+
+# Wait for keyboard input to continue or exit
+while True:
+    user_input = input("Press Enter to continue or '0' to exit: ")
+    if user_input == '0':
+        print("Exiting simulation...")
+        exit()
+    elif user_input == '':
+        print("Continuing simulation...")
+        break
+    else:
+        print("Invalid input. Please press Enter to continue or '0' to exit.")
+
+dt = 1e-6#particles.compute_dt()
+
+print(f"dt: {dt}")
+# Create MPM solver
+
+solver = MPMSolver(particles, grid, dt)
+
+
+
+num_steps = 10000  # TODO: Adjust as needed
+
+
+
+# Create an output directory if it doesn't exist
+output_dir = "simulation_output"
+os.makedirs(output_dir, exist_ok=True)
+
+# Main simulation loop
+for step in tqdm(range(num_steps), desc="Simulating"):
+    solver.step()
+    
+    # Save outputs every 100 steps
+    if step % 1000 == 0:
+        output_file = os.path.join(output_dir, f"step_{step:05d}.csv")
+        save_particles_to_csv(particles, output_file)
+        print(f"Saved output for step {step} to {output_file}")
+
+print("Simulation completed successfully!")
diff --git a/material.py b/material.py
new file mode 100644
index 0000000..536aaa3
--- /dev/null
+++ b/material.py
@@ -0,0 +1,27 @@
+import numpy as np
+
+class LinearElastic:
+    def __init__(self, density, youngs_modulus, poisson_ratio):
+        self.density = density
+        self.youngs_modulus = youngs_modulus
+        self.poisson_ratio = poisson_ratio
+        
+        # Compute Lame parameters
+        self.lambda_ = (youngs_modulus * poisson_ratio) / ((1 + poisson_ratio) * (1 - 2 * poisson_ratio))
+        self.mu = youngs_modulus / (2 * (1 + poisson_ratio))
+
+    def compute_stress_rate(self, strain_rate):
+        # Compute stress rate using Hooke's law for 2D stress state
+        # Expect strain_rate as a vector [exx, eyy, exy]
+        assert strain_rate.shape == (2,2), "Strain rate must be a 2x2 matrix"
+        
+        exx, eyy, exy, eyx = strain_rate.flatten()
+        
+        # Compute stress components
+        factor = self.youngs_modulus / ((1 + self.poisson_ratio) * (1 - 2 * self.poisson_ratio))
+        sxx = factor * ((1 - self.poisson_ratio) * exx + self.poisson_ratio * (eyy))
+        syy = factor * ((1 - self.poisson_ratio) * eyy + self.poisson_ratio * (exx))
+        sxy = self.mu * 2 * exy
+        
+        return np.array([[sxx, sxy], [sxy, syy]])
+
diff --git a/nodes.py b/nodes.py
new file mode 100644
index 0000000..bf3b05e
--- /dev/null
+++ b/nodes.py
@@ -0,0 +1,57 @@
+import numpy as np
+
+class Grid:
+    def __init__(self, physical_size, cell_size):
+        self.physical_size = np.array(physical_size)
+        self.cell_size = cell_size
+        self.node_count = np.ceil(self.physical_size / self.cell_size).astype(int) + 1
+        self.nodes = self.initialize_grid()
+        self.total_nodes = np.prod(self.node_count)
+
+    def initialize_grid(self):
+        grid = np.empty(self.node_count, dtype=object)
+        for i in range(self.node_count[0]):
+            for j in range(self.node_count[1]):
+                x = i * self.cell_size
+                y = j * self.cell_size
+                grid[i, j] = Node(position=np.array([x, y]))
+        return grid
+
+    def reset_nodes(self):
+        for i in range(self.node_count[0]):
+            for j in range(self.node_count[1]):
+                self.nodes[i, j].reset()
+
+    def get_node(self, i, j):
+        return self.nodes[i, j]
+
+    def set_node(self, i, j, node):
+        self.nodes[i, j] = node
+    
+    def get_nearby_nodes(self, particle_position):
+        i = int(particle_position[0] / self.cell_size)
+        j = int(particle_position[1] / self.cell_size)
+        nearby_nodes = []
+        for di in range(-1, 2):
+            for dj in range(-1, 2):
+                if 0 <= i + di < self.node_count[0] and 0 <= j + dj < self.node_count[1]:
+                    nearby_nodes.append(self.nodes[i + di, j + dj])
+        return nearby_nodes
+
+    @property
+    def node_count_total(self):
+        return self.total_nodes
+
+class Node:
+    def __init__(self, position):
+        self.position = position
+        self.mass = 0.0
+        self.velocity = np.zeros(2)
+        self.momentum = np.zeros(2)
+        self.force = np.zeros(2)
+
+    def reset(self):
+        self.mass = 0.0
+        self.velocity.fill(0)
+        self.force.fill(0)
+        # Reset other properties as needed
\ No newline at end of file
diff --git a/particle_shapes.py b/particle_shapes.py
new file mode 100644
index 0000000..63df188
--- /dev/null
+++ b/particle_shapes.py
@@ -0,0 +1,97 @@
+import numpy as np
+
+class ParticleShape:
+    def __init__(self, cell_size, object_id):
+        self.cell_size = cell_size
+        self.particles_per_cell = 4  # 4 particles per cell in 2D
+        self.object_id = object_id
+
+    def generate_particles(self):
+        raise NotImplementedError("Subclasses must implement generate_particles method")
+
+    def _generate_grid_particles(self, x_range, y_range):
+        particles = []
+        for i in range(x_range[0], x_range[1]):
+            for j in range(y_range[0], y_range[1]):
+                for px in range(2):
+                    for py in range(2):
+                        # Place particles at 1/4 and 3/4 of the cell in each dimension
+                        x = (i + (px + 1) / 3) * self.cell_size
+                        y = (j + (py + 1) / 3) * self.cell_size
+                        
+                        if self._is_inside(x, y):
+                            particle = {
+                                'position': np.array([x, y]),
+                                'velocity': np.zeros(2),
+                                'mass': self.density * self.cell_size**2 / self.particles_per_cell,
+                                'volume': self.cell_size**2 / self.particles_per_cell,
+                                'stress': np.zeros((2, 2)),
+                                'strain': np.zeros((2, 2)),
+                                'object_id': self.object_id,  # Add object_id to the particle
+                            }
+                            particles.append(particle)
+        return particles
+
+    def _is_inside(self, x, y):
+        raise NotImplementedError("Subclasses must implement _is_inside method")
+
+class Block(ParticleShape):
+    def __init__(self, cell_size, width, height, position, object_id):
+        super().__init__(cell_size, object_id)
+        self.width = width
+        self.height = height
+        self.position = np.array(position)
+
+    def generate_particles(self):
+        num_cells_x = int(self.width / self.cell_size)
+        num_cells_y = int(self.height / self.cell_size)
+        
+        x_range = (0, num_cells_x)
+        y_range = (0, num_cells_y)
+        
+        return self._generate_grid_particles(x_range, y_range)
+
+    def _is_inside(self, x, y):
+        return (0 <= x < self.width) and (0 <= y < self.height)
+
+class Disk(ParticleShape):
+    def __init__(self, cell_size, radius, center, object_id):
+        super().__init__(cell_size, object_id)
+        self.radius = radius
+        self.center = np.array(center)
+        print(f"Disk created with radius={radius}, center={center}")
+
+    def generate_particles(self):
+        particles = []
+        x_min, x_max = self.center[0] - self.radius, self.center[0] + self.radius
+        y_min, y_max = self.center[1] - self.radius, self.center[1] + self.radius
+        
+        for i in range(int(x_min / self.cell_size), int(x_max / self.cell_size) + 1):
+            for j in range(int(y_min / self.cell_size), int(y_max / self.cell_size) + 1):
+                for px in range(2):
+                    for py in range(2):
+                        x = (i + (px + 1) / 3) * self.cell_size
+                        y = (j + (py + 1) / 3) * self.cell_size
+                        if self._is_inside(x, y):
+                            particle = {
+                                'position': np.array([x, y]),
+                                'velocity': np.zeros(2),
+                                'mass': 1.0 ,
+                                'volume': 1.0,
+                                'stress': np.zeros((2, 2)),
+                                'strain': np.zeros((2, 2)),
+                                'object_id': self.object_id,  # Add object_id to the particle
+                            }
+                            particles.append(particle)
+        
+        print(f"Disk generated {len(particles)} particles")
+        if len(particles) == 0:
+            print(f"Warning: No particles generated. Check disk parameters.")
+        return particles
+
+    def _is_inside(self, x, y):
+        dx = x - self.center[0]
+        dy = y - self.center[1]
+        return dx**2 + dy**2 <= self.radius**2
+
+
diff --git a/particles.py b/particles.py
new file mode 100644
index 0000000..f6124a8
--- /dev/null
+++ b/particles.py
@@ -0,0 +1,115 @@
+import numpy as np
+from material import LinearElastic  # Import your material models
+
+class Particles:
+    def __init__(self, particle_data, material_properties):
+        self.particles = []
+        self.material_properties = material_properties
+        print(f"Initializing Particles with {len(particle_data)} particle data")
+        self.create_particles(particle_data)
+        print(f"Finished creating {len(self.particles)} particles")
+
+    def create_particles(self, particle_data):
+        for i, data in enumerate(particle_data):
+            try:
+                position = np.array(data['position'], dtype=float)
+                velocity = np.array(data.get('velocity', [0.0, 0.0]), dtype=float)
+                
+                # Ensure position and velocity are 2D vectors
+                if position.shape != (2,):
+                    raise ValueError(f"Position must be a 2D vector. Got shape {position.shape}")
+                if velocity.shape != (2,):
+                    raise ValueError(f"Velocity must be a 2D vector. Got shape {velocity.shape}")
+
+                object_id = data['object_id']
+                material_props = self.material_properties[object_id]
+
+                particle = {
+                    'position': position,
+                    'velocity': velocity,
+                    'density': material_props['density'],
+                    'mass': data['mass'],
+                    'volume': data['volume'],
+                    'stress': np.zeros((2, 2)),
+                    'strain': np.zeros((2, 2)),
+                    'id': data.get('id', len(self.particles)),
+                    'object_id': object_id,
+                    'material': self.create_material(material_props)
+                }
+                self.particles.append(particle)
+            except Exception as e:
+                print(f"Error creating particle {i}: {str(e)}")
+                print(f"Particle data: {data}")
+                print(f"Material properties: {material_props}")
+
+    def create_material(self, properties):
+        if properties['type'] == "LinearElastic":
+            return LinearElastic(
+                youngs_modulus=properties['youngs_modulus'],
+                poisson_ratio=properties['poisson_ratio'],
+                density=properties['density']
+            )
+        else:
+            raise ValueError(f"Unknown material type: {properties['type']}")
+
+    def get_particle_count(self):
+        return len(self.particles)
+
+    def assign_material(self, material_type, properties):
+        material = {
+            "type": material_type,
+            "properties": properties
+        }
+        return material
+
+    def get_particles(self):
+        return self.particles
+    
+    def get_particle(self, index):
+        return self.particles[index]    
+    
+    def compute_dt(self):
+        # Compute the time step using CFL condition with stiffness
+        cfl_factor = 0.2
+        max_wave_speed = 0.0
+        min_cell_size = float('inf')
+
+        for particle in self.particles:
+            # Calculate the wave speed for each particle
+            if 'material' in particle and 'properties' in particle['material']:
+                properties = particle['material']['properties']
+                if 'youngs_modulus' in properties and 'density' in properties:
+                    E = properties['youngs_modulus']
+                    rho = properties['density']
+                    wave_speed = np.sqrt(E / rho)
+                    max_wave_speed = max(max_wave_speed, wave_speed)
+
+            # Find the minimum cell size
+            if 'cell_size' in particle:
+                min_cell_size = min(min_cell_size, particle['cell_size'])
+
+        # If cell_size is not available in particle data, use a default value
+        if min_cell_size == float('inf'):
+            min_cell_size = 0.05  # Default value, adjust as needed
+
+        # Compute dt using CFL condition with stiffness
+        if max_wave_speed > 0:
+            dt = cfl_factor * min_cell_size / max_wave_speed
+        else:
+            # Fallback if no valid wave speed could be calculated
+            dt = cfl_factor * min_cell_size
+
+        return dt
+    
+    def set_object_velocity(particles, object_id, velocity):
+        for particle in particles.particles:
+            if particle['object_id'] == object_id:
+                particle['velocity'] = np.array(velocity)
+
+    def assign_material_properties(self, material, object_id):
+        for particle in self.particles:
+            if particle['object_id'] == object_id:
+                particle['material'] = self.create_material(material)
+                particle['density'] = material['density']
+                particle['volume'] = particle['mass'] / particle['density']
+
diff --git a/plotting.py b/plotting.py
new file mode 100644
index 0000000..dd77a81
--- /dev/null
+++ b/plotting.py
@@ -0,0 +1,98 @@
+import matplotlib.pyplot as plt
+import matplotlib.patches as patches
+import numpy as np
+
+def visualize_particles_and_grid(particles, grid, step):
+    fig, ax = plt.subplots(figsize=(5, 5))
+    
+    # Extract positions and velocities
+    positions = np.array([p['position'] for p in particles.get_particles()])
+    velocities = np.array([p['velocity'] for p in particles.get_particles()])
+    
+    # Ensure velocities is 2D
+    if velocities.ndim == 1:
+        velocities = velocities.reshape(-1, 1)
+    
+    # Calculate velocity magnitudes
+    velocity_magnitudes = np.linalg.norm(velocities, axis=1)
+    
+    # Create color map for particles
+    cmap = plt.get_cmap('viridis')
+    norm = plt.Normalize(vmin=velocity_magnitudes.min(), vmax=velocity_magnitudes.max())
+    
+    # Draw particles as squares
+    square_size = grid.cell_size / 2
+    for pos, vel_mag in zip(positions, velocity_magnitudes):
+        color = cmap(norm(vel_mag))
+        square = patches.Rectangle((pos[0] - square_size/2, pos[1] - square_size/2), 
+                                   square_size, square_size, 
+                                   facecolor=color, edgecolor='none', alpha=0.8)
+        ax.add_patch(square)
+    
+    # Draw MPM grid
+    for i in range(grid.node_count[0] + 1):
+        ax.axvline(x=i * grid.cell_size, color='gray', linestyle=':', alpha=0.5)
+    for j in range(grid.node_count[1] + 1):
+        ax.axhline(y=j * grid.cell_size, color='gray', linestyle=':', alpha=0.5)
+    
+    # Set plot properties
+    ax.set_aspect('equal')
+    ax.set_xlim(0, grid.physical_size[0])
+    ax.set_ylim(0, grid.physical_size[1])
+    ax.set_xlabel('X (m)')
+    ax.set_ylabel('Y (m)')
+    ax.set_title(f'Particles and MPM Grid - Step {step}')
+    
+    # Add colorbar
+    sm = plt.cm.ScalarMappable(cmap=cmap, norm=norm)
+    sm.set_array([])
+    cbar = fig.colorbar(sm, ax=ax, label='Velocity Magnitude')
+    
+    plt.tight_layout()
+    plt.show()
+
+def visualize_particles_with_object_id(particles, grid):
+    fig, ax = plt.subplots(figsize=(5, 5))
+    # Extract positions and object IDs
+    positions = np.array([p['position'] for p in particles.get_particles()])
+    object_ids = np.array([p['object_id'] for p in particles.get_particles()])
+    
+    print(f"Total particles: {len(positions)}")
+    print(f"Unique object IDs: {np.unique(object_ids)}")
+    # print(f"Position range: x({positions[:, 0].min()}, {positions[:, 0].max()}), y({positions[:, 1].min()}, {positions[:, 1].max()})")
+    
+    # Create a color map for different object IDs
+    unique_ids = np.unique(object_ids)
+    colors = plt.cm.rainbow(np.linspace(0, 1, len(unique_ids)))
+    color_map = dict(zip(unique_ids, colors))
+    
+    # Draw particles as squares
+    square_size = grid.cell_size / 2
+    for pos, obj_id in zip(positions, object_ids):
+        color = color_map[obj_id]
+        square = patches.Rectangle((pos[0] - square_size/2, pos[1] - square_size/2), 
+                                   square_size, square_size, 
+                                   facecolor=color, edgecolor='none', alpha=0.8)
+        ax.add_patch(square)
+    
+    # Draw MPM grid
+    for i in range(grid.node_count[0] + 1):
+        ax.axvline(x=i * grid.cell_size, color='gray', linestyle=':', alpha=0.5)
+    for j in range(grid.node_count[1] + 1):
+        ax.axhline(y=j * grid.cell_size, color='gray', linestyle=':', alpha=0.5)
+    
+    # Set plot properties
+    ax.set_aspect('equal')
+    ax.set_xlim(0, grid.physical_size[0])
+    ax.set_ylim(0, grid.physical_size[1])
+    ax.set_xlabel('X (m)')
+    ax.set_ylabel('Y (m)')
+    ax.set_title('Particles with Object ID')
+    
+    # Add legend
+    legend_elements = [patches.Patch(facecolor=color_map[id], edgecolor='none', label=f'Object {id}') 
+                       for id in unique_ids]
+    ax.legend(handles=legend_elements, loc='upper right')
+    
+    plt.tight_layout()
+    plt.show()
\ No newline at end of file
diff --git a/results.py b/results.py
new file mode 100644
index 0000000..c6823a8
--- /dev/null
+++ b/results.py
@@ -0,0 +1,38 @@
+import csv
+import numpy as np
+
+def save_particles_to_csv(particles, output_file):
+    with open(output_file, 'w', newline='') as csvfile:
+        csv_writer = csv.writer(csvfile)
+        # Write header
+        csv_writer.writerow([
+            'particle_id', 'object_id', 'pos_x', 'pos_y', 'vel_x', 'vel_y',
+            'stress_xx', 'stress_xy', 'stress_yx', 'stress_yy',
+            'strain_xx', 'strain_xy', 'strain_yx', 'strain_yy',
+            'volume', 'mass', 'density',
+            'youngs_modulus', 'poisson_ratio'
+        ])
+        
+        # Write particle data
+        for p_idx in range(particles.get_particle_count()):
+            particle = particles.get_particle(p_idx)
+            pos = particle['position']
+            vel = particle['velocity']
+            stress = particle['stress']
+            strain = particle['strain']
+            material = particle['material']
+            
+            csv_writer.writerow([
+                p_idx,
+                particle['object_id'],
+                pos[0], pos[1],
+                vel[0], vel[1],
+                stress[0,0], stress[0,1], stress[1,0], stress[1,1],
+                strain[0,0], strain[0,1], strain[1,0], strain[1,1],
+                particle['volume'],
+                particle['mass'],
+                particle['density'],
+                material.youngs_modulus,
+                material.poisson_ratio
+            ])
+
diff --git a/results_vis.py b/results_vis.py
new file mode 100644
index 0000000..b2d45a2
--- /dev/null
+++ b/results_vis.py
@@ -0,0 +1,179 @@
+import os
+import csv
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.animation import FuncAnimation
+from matplotlib.colors import Normalize
+
+# Directory containing the CSV files
+data_dir = "simulation_output"
+
+# Get all CSV files in the directory
+csv_files = sorted([f for f in os.listdir(data_dir) if f.endswith('.csv')])
+
+# Variables to store grid boundaries
+grid_min_x = float('inf')
+grid_max_x = float('-inf')
+grid_min_y = float('inf')
+grid_max_y = float('-inf')
+
+# Function to read a CSV file
+def read_csv(filename):
+    global grid_min_x, grid_max_x, grid_min_y, grid_max_y
+    with open(os.path.join(data_dir, filename), 'r') as f:
+        reader = csv.reader(f)
+        header = next(reader)  # Read the header
+        data = [list(map(float, row)) for row in reader if row]  # Convert to float and skip empty rows
+    if not data:
+        return np.array([]), np.array([]), np.array([]), np.array([]), np.array([])
+    data = np.array(data)
+    positions = data[:, 2:4]
+    velocities = data[:, 4:6]
+    stress = data[:, 6:10]
+    volumes = data[:, 14]
+    masses = data[:, 15]
+    
+    # Update grid boundaries
+    grid_min_x = min(grid_min_x, positions[:, 0].min())
+    grid_max_x = max(grid_max_x, positions[:, 0].max())
+    grid_min_y = min(grid_min_y, positions[:, 1].min())
+    grid_max_y = max(grid_max_y, positions[:, 1].max())
+    
+    return positions, velocities, stress, volumes, masses
+
+# Read the first file to get the number of particles and initialize the plot
+initial_positions, initial_velocities, initial_stress, initial_volumes, initial_masses = read_csv(csv_files[0])
+num_particles = len(initial_positions)
+
+# Create the figure and axes
+fig = plt.figure(figsize=(20, 15))
+gs = fig.add_gridspec(2, 2)
+ax1 = fig.add_subplot(gs[0, 0])
+ax2 = fig.add_subplot(gs[0, 1])
+ax3 = fig.add_subplot(gs[1, :])
+
+scatter1 = ax1.scatter([], [], s=10, c=[], cmap='viridis')
+scatter2 = ax2.scatter([], [], s=10, c=[], cmap='plasma')
+line1, = ax3.plot([], [], label='Kinetic Energy')
+line2, = ax3.plot([], [], label='Elastic Energy')
+line3, = ax3.plot([], [], label='Total Energy')
+
+ax1.set_aspect('equal')
+ax2.set_aspect('equal')
+ax1.set_title("Velocity Magnitude")
+ax2.set_title("Von Mises Stress")
+ax3.set_title("Energy Evolution")
+ax3.set_xlabel("Step")
+ax3.set_ylabel("Energy")
+ax3.legend()
+
+title = fig.suptitle("Step: 0")
+
+# Add colorbars
+cbar1 = plt.colorbar(scatter1, ax=ax1)
+cbar2 = plt.colorbar(scatter2, ax=ax2)
+cbar1.set_label('Velocity magnitude')
+cbar2.set_label('Von Mises stress')
+
+# Initialize the color normalizations
+norm1 = Normalize()
+norm2 = Normalize()
+
+# Lists to store energy values and stress range
+kinetic_energy = []
+elastic_energy = []
+total_energy = []
+stress_min = float('inf')
+stress_max = float('-inf')
+
+# Function to calculate von Mises stress
+def von_mises_stress(stress):
+    xx, xy, yx, yy = stress.T
+    return np.sqrt(0.5 * ((xx - yy)**2 + (yy - xx)**2 + 6 * (xy**2)))
+
+# Function to calculate kinetic energy
+def calculate_kinetic_energy(velocities, masses):
+    return 0.5 * np.sum(masses[:, np.newaxis] * np.sum(velocities**2, axis=1))
+
+# Function to calculate elastic energy
+def calculate_elastic_energy(stress, volumes, youngs_modulus):
+    return 0.5 * np.sum(volumes[:, np.newaxis] * np.sum(stress**2, axis=1)) / youngs_modulus
+
+# Animation update function
+def update(frame):
+    global kinetic_energy, elastic_energy, total_energy, stress_min, stress_max
+    
+    # Reset energy lists and stress range if the animation is starting over
+    if frame == 0:
+        kinetic_energy = []
+        elastic_energy = []
+        total_energy = []
+        stress_min = float('inf')
+        stress_max = float('-inf')
+    
+    positions, velocities, stress, volumes, masses = read_csv(csv_files[frame])
+    if len(positions) == 0:
+        return scatter1, scatter2, line1, line2, line3, title
+    
+    scatter1.set_offsets(positions)
+    scatter2.set_offsets(positions)
+    
+    # Calculate velocity magnitudes
+    vel_mag = np.linalg.norm(velocities, axis=1)
+    
+    # Calculate von Mises stress
+    von_mises = von_mises_stress(stress)
+    
+    # Update stress range
+    stress_min = min(stress_min, von_mises.min())
+    stress_max = max(stress_max, von_mises.max())
+    
+    # Update color normalizations
+    norm1.autoscale(vel_mag)
+    norm2.vmin = stress_min
+    norm2.vmax = stress_max
+    
+    scatter1.set_array(vel_mag)
+    scatter2.set_array(von_mises)
+    
+    # Update colorbars
+    scatter1.set_norm(norm1)
+    scatter2.set_norm(norm2)
+    cbar1.update_normal(scatter1)
+    cbar2.update_normal(scatter2)
+    
+    # Calculate and store energies
+    youngs_modulus = float(csv_files[frame].split('_')[-1].split('.')[0])  # Extract Young's modulus from filename
+    ke = calculate_kinetic_energy(velocities, masses)
+    ee = calculate_elastic_energy(stress, volumes, youngs_modulus)
+    te = ke + ee
+    kinetic_energy.append(ke)
+    elastic_energy.append(ee)
+    total_energy.append(te)
+    
+    # Update energy plots
+    steps = range(len(kinetic_energy))
+    line1.set_data(steps, kinetic_energy)
+    line2.set_data(steps, elastic_energy)
+    line3.set_data(steps, total_energy)
+    ax3.relim()
+    ax3.autoscale_view()
+    
+    # Update axes limits
+    ax1.set_xlim(grid_min_x, grid_max_x)
+    ax1.set_ylim(grid_min_y, grid_max_y)
+    ax2.set_xlim(grid_min_x, grid_max_x)
+    ax2.set_ylim(grid_min_y, grid_max_y)
+    
+    title.set_text(f"Step: {frame}")
+    return scatter1, scatter2, line1, line2, line3, title
+
+# Create the animation
+anim = FuncAnimation(fig, update, frames=len(csv_files), interval=500, blit=False)
+
+# Save the animation (optional)
+# anim.save('simulation.mp4', writer='ffmpeg', fps=30)
+
+# Show the plot
+plt.tight_layout()
+plt.show()
\ No newline at end of file
diff --git a/shape_function.py b/shape_function.py
new file mode 100644
index 0000000..56e2c86
--- /dev/null
+++ b/shape_function.py
@@ -0,0 +1,42 @@
+import numpy as np
+
+def shape_function(particle_pos, node_pos, cell_size):
+    """
+    Calculate the linear nodal shape function value (tent function).
+    
+    Args:
+    particle_pos (np.array): Position of the particle (2D)
+    node_pos (np.array): Position of the grid node (2D)
+    cell_size (float): Size of a grid cell
+    
+    Returns:
+    tuple: Shape function values (sx, sy)
+    """
+    dx = (particle_pos[0] - node_pos[0]) / cell_size
+    dy = (particle_pos[1] - node_pos[1]) / cell_size
+    
+    sx = 1 - abs(dx) if abs(dx) < 1 else 0
+    sy = 1 - abs(dy) if abs(dy) < 1 else 0
+    
+    return sx, sy
+
+def gradient_shape_function(particle_pos, node_pos, cell_size):
+    """
+    Calculate the gradient of the linear nodal shape function.
+    
+    Args:
+    particle_pos (np.array): Position of the particle (2D)
+    node_pos (np.array): Position of the grid node (2D)
+    cell_size (float): Size of a grid cell
+    
+    Returns:
+    tuple: Gradient of the shape function (dsx_dx, dsy_dy)
+    """
+    dx = (particle_pos[0] - node_pos[0]) / cell_size
+    dy = (particle_pos[1] - node_pos[1]) / cell_size
+    
+    dsx_dx = -1 if -1 < dx < 0 else (1 if 0 <= dx < 1 else 0)
+    dsy_dy = -1 if -1 < dy < 0 else (1 if 0 <= dy < 1 else 0)
+    
+    return dsx_dx / cell_size, dsy_dy / cell_size
+
diff --git a/simulation_output/.DS_Store b/simulation_output/.DS_Store
new file mode 100644
index 0000000..5008ddf
Binary files /dev/null and b/simulation_output/.DS_Store differ
diff --git a/simulation_output/step_00000.csv b/simulation_output/step_00000.csv
new file mode 100644
index 0000000..f3ff238
--- /dev/null
+++ b/simulation_output/step_00000.csv
@@ -0,0 +1,1609 @@
+particle_id,object_id,pos_x,pos_y,vel_x,vel_y,stress_xx,stress_xy,stress_yx,stress_yy,strain_xx,strain_xy,strain_yx,strain_yy,volume,mass,density,youngs_modulus,poisson_ratio
+0,1,0.12666866666666665,0.4733333333333334,2.0000000000000075,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+1,1,0.12666866666666665,0.4766666666666666,2.0000000000000075,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+2,1,0.12333533333333334,0.4833333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+3,1,0.12333533333333334,0.48666666666666664,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+4,1,0.12666866666666665,0.4833333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+5,1,0.12666866666666665,0.48666666666666664,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+6,1,0.12333533333333334,0.49333333333333335,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+7,1,0.12333533333333334,0.49666666666666665,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+8,1,0.12666866666666665,0.49333333333333335,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+9,1,0.12666866666666665,0.49666666666666665,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+10,1,0.12333533333333334,0.5033333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+11,1,0.12333533333333334,0.5066666666666666,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+12,1,0.12666866666666665,0.5033333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+13,1,0.12666866666666665,0.5066666666666666,1.9999999999999962,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+14,1,0.12333533333333334,0.5133333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+15,1,0.12333533333333334,0.5166666666666666,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+16,1,0.12666866666666665,0.5133333333333334,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+17,1,0.12666866666666665,0.5166666666666666,1.9999999999999962,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
+18,1,0.12666866666666665,0.5233333333333333,1.9999999999999967,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
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+20,1,0.13333533333333333,0.45666666666666667,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
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+22,1,0.13666866666666666,0.45666666666666667,1.9999999999999964,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,0.0,1.0,1.0,1000.0,1000000.0,0.3
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diff --git a/simulation_output/step_01000.csv b/simulation_output/step_01000.csv
new file mode 100644
index 0000000..0e4a92f
--- /dev/null
+++ b/simulation_output/step_01000.csv
@@ -0,0 +1,1609 @@
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diff --git a/solver.py b/solver.py
new file mode 100644
index 0000000..e7b3893
--- /dev/null
+++ b/solver.py
@@ -0,0 +1,170 @@
+import numpy as np
+from particles import Particles
+from nodes import Grid
+from shape_function import shape_function, gradient_shape_function
+
+
+class MPMSolver:
+    def __init__(self, particles: Particles, grid: Grid, dt: float):
+        self.particles = particles
+        self.grid = grid
+        self.dt = dt
+        self.time = 0.0
+        
+        # Initialize initial_volume and initial_density for each particle
+        for particle in self.particles.particles:
+            if 'initial_volume' not in particle:
+                particle['initial_volume'] = particle['volume']
+            if 'initial_density' not in particle:
+                particle['initial_density'] = particle['density']
+
+    def step(self):
+        self.prepare_step()
+        self.particle_to_grid()
+        self.update_grid()
+        self.grid_to_particle()
+        self.update_particles()
+        # self.apply_boundary_conditions()
+        self.time += self.dt
+
+    def prepare_step(self):
+        self.grid.reset_nodes()
+        
+        # Initialize node_particle_map as a 2D list of lists
+        self.grid.node_particle_map = [[[] for _ in range(self.grid.node_count[1])] 
+                                       for _ in range(self.grid.node_count[0])]
+
+        for p_idx in range(len(self.particles.particles)):
+            particle_pos = self.particles.particles[p_idx]['position']
+            i = int(particle_pos[0] / self.grid.cell_size)
+            j = int(particle_pos[1] / self.grid.cell_size)
+
+            node_range = 2  # Influence radius
+
+            for di in range(-node_range, node_range + 1):
+                for dj in range(-node_range, node_range + 1):
+                    node_i = i + di
+                    node_j = j + dj
+
+                    if 0 <= node_i < self.grid.node_count[0] and 0 <= node_j < self.grid.node_count[1]:
+                        node_pos = np.array([node_i * self.grid.cell_size, node_j * self.grid.cell_size])
+                        shape_x, shape_y = shape_function(particle_pos, node_pos, self.grid.cell_size)
+
+                        if shape_x * shape_y > 0:
+                            self.grid.node_particle_map[node_i][node_j].append(p_idx)
+
+    def particle_to_grid(self):
+        for i in range(self.grid.node_count[0]):
+            for j in range(self.grid.node_count[1]):
+                node = self.grid.nodes[i, j]
+                node_pos = np.array([i * self.grid.cell_size, j * self.grid.cell_size])
+
+                node.force = np.zeros(2)
+                node.mass = 0
+                node.momentum = np.zeros(2)
+
+                particle_indices = self.grid.node_particle_map[i][j]
+
+                for p_idx in particle_indices:
+                    particle = self.particles.get_particle(p_idx)
+                    particle_pos = particle['position']
+
+                    shape_x, shape_y = shape_function(particle_pos, node_pos, self.grid.cell_size)
+                    grad_shape_x, grad_shape_y = gradient_shape_function(particle_pos, node_pos, self.grid.cell_size)
+
+                    shape_value = shape_x * shape_y
+
+                    # Update node properties
+                    node.mass += particle['mass'] * shape_value
+                    node.momentum += particle['mass'] * particle['velocity'] * shape_value
+
+                    force_x = -(particle['mass'] / particle['density']) * (
+                        grad_shape_x * shape_y * particle['stress'][0, 0] + 
+                        grad_shape_y * shape_x * particle['stress'][0, 1]
+                    )
+                    force_y = -(particle['mass'] / particle['density']) * (
+                        grad_shape_x * shape_y * particle['stress'][1, 0] + 
+                        grad_shape_y * shape_x * particle['stress'][1, 1]
+                    )
+
+                    node.force[0] += force_x
+                    node.force[1] += force_y
+
+                if node.mass > 0:
+                    node.velocity = node.momentum / node.mass
+                else:
+                    node.velocity = np.zeros(2)
+
+    def update_grid(self):
+        dt = self.dt  # Time step
+        for node in self.grid.nodes.flat:
+            if node.mass > 1e-9:
+                # Update momentum using explicit integration
+                node.momentum += node.force * dt
+
+                # Update velocity based on new momentum
+                node.velocity = node.momentum / node.mass
+
+        # Apply external forces (e.g., gravity) if needed
+        gravity = np.array([0, 0]) 
+        for node in self.grid.nodes.flat:
+            if node.mass > 0:
+                node.velocity += gravity * dt
+
+        # Clear forces for the next step
+        for node in self.grid.nodes.flat:
+            node.force = np.zeros(2)
+
+    def grid_to_particle(self):
+      for p_idx, particle in enumerate(self.particles.particles):
+            particle_velocity_update = np.zeros(2)
+            strain_rate = np.zeros((2, 2))
+
+            nearby_nodes = self.grid.get_nearby_nodes(particle['position'])
+
+            for node in nearby_nodes:
+                if node.mass > 1e-9:
+                    shape_x, shape_y = shape_function(particle['position'], node.position, self.grid.cell_size)
+                    grad_shape_x, grad_shape_y = gradient_shape_function(particle['position'], node.position, self.grid.cell_size)
+
+                    shape_value = shape_x * shape_y
+                    particle_velocity_update += shape_value * node.velocity
+
+                    # Calculate strain rate components
+                    strain_rate[0, 0] += grad_shape_x * shape_y * node.velocity[0]  # exx
+                    strain_rate[0, 1] += 0.5 * (grad_shape_x * shape_y * node.velocity[1] + 
+                                                grad_shape_y * shape_x * node.velocity[0])  # exy
+                    strain_rate[1, 0] = strain_rate[0, 1]  # eyx = exy
+                    strain_rate[1, 1] += grad_shape_y * shape_x * node.velocity[1]  # eyy
+
+            # Update particle velocity
+            particle['velocity'] = particle_velocity_update
+
+            # Store strain rate in particle
+            particle['strain_rate'] = strain_rate
+
+    def update_particles(self):
+        for particle in self.particles.particles:
+            # Update position using current velocity
+            particle['position'] += particle['velocity'] * self.dt
+
+            # Update deformation gradient (F)
+            identity = np.eye(2)
+            particle['F'] = np.dot(identity + self.dt * particle['strain_rate'], particle.get('F', identity))
+
+            # Update volume
+            J = np.linalg.det(particle['F'])
+            particle['volume'] = particle['initial_volume'] * J
+
+            # Update density
+            particle['density'] = particle['initial_density'] / J
+
+            # Update stress using the material model
+            material = particle['material']
+            stress_rate = material.compute_stress_rate(particle['strain_rate'])
+            particle['stress'] += stress_rate * self.dt
+
+    def apply_boundary_conditions(self):
+        pass
+        # Implement boundary conditions here
+