Inspired by my son Ridit’s enthusiasm for computer graphic and being informed by him about Incremental Potential Contact (IPC), I’ve been diving into this cutting-edge technique that addresses a persistent challenge in physics simulations: preventing objects from penetrating each other. Introduced in a 2020 SIGGRAPH paper by Li et al., IPC has become a game-changer for achieving realistic, stable, and penetration-free simulations in computer graphics and beyond. In this blog post, I’ll explain what IPC is, how it tackles penetration, and why it’s a significant advancement for real-time applications like video games, animations, and virtual reality.The Penetration Problem in Physics SimulationsIn physics-based simulations, objects often need to collide, slide, or rest against each other realistically. Traditional methods, like those used in game engines (e.g., Bullet Physics in Blender) or finite element simulations, rely on discrete collision detection and penalty-based forces. These approaches check for overlaps at fixed time steps and apply repulsive forces to push objects apart. However, they have limitations:
Key features of IPC include:
m and stiffness N/m. The script computes the barrier energy and contact forces to prevent penetration:
Here's my experimentation of IPC in Python
- Penetration Artifacts: Objects can pass through each other, especially at high velocities or with small time steps, leading to visual glitches (e.g., a character’s arm clipping through a wall).
- Numerical Instability: Penalty methods can cause jittering or oscillations, particularly in stiff systems like cloth or stacked objects.
- Complex Geometries: Deformable objects (e.g., soft bodies, hair) or thin surfaces often exacerbate penetration issues, requiring costly post-processing to fix.
Key features of IPC include:
- Barrier Potential: A mathematical function that increases as objects approach, preventing them from getting closer than a minimum distance.
- Incremental Solving: Contact constraints are solved iteratively in small steps, ensuring stability even for large time steps or complex scenes.
- Friction and Elasticity: IPC naturally incorporates friction and material properties, enabling realistic sliding and bouncing.
- Contact Detection: IPC identifies potential contact points (e.g., vertex-vertex, vertex-edge, or edge-edge pairs) using proximity queries on a collision mesh. This is done continuously, not just at discrete time steps.
- Barrier Potential: For each potential contact, IPC defines a barrier energy ( B(d) ), typically a logarithmic or polynomial function that grows as the distance ( d ) between objects approaches zero. For example:Here,
B(d) = -\kappa \log(d), for d > 0is a stiffness parameter, and the energy becomes infinite as\kappa, ensuring no penetration occurs.d tends to 0 - Incremental Optimization: IPC minimizes the total system energy (including kinetic, elastic, and barrier energies) subject to contact constraints. It uses an iterative solver (e.g., Newton’s method or projected gradient descent) to adjust object positions and velocities incrementally, resolving contacts step-by-step.
- Time Integration: IPC employs implicit time integration to handle stiff systems, allowing larger time steps without sacrificing stability. This is crucial for real-time applications where performance is paramount.
- Guaranteed Non-Penetration: The barrier potential ensures objects maintain a minimum separation, eliminating clipping artifacts.
- Stability: Incremental solving and implicit integration handle stiff systems (e.g., rigid bodies or thin cloth) without jittering or divergence.
- Versatility: IPC excels with complex geometries, including deformable objects, thin shells, and codimensional structures (e.g., hair or ropes).
- Real-Time Potential: Optimized implementations, like the IPC Toolkit, leverage GPU parallelization, making IPC viable for games and VR.
d_min = 0.01kappa = 10^5- Distance Calculation: The distance between particles is 0.005 m, violating .
d_min - Barrier Energy: A quadratic barrier yields E = 0.5 * kappa (d_min - d)^2 = 1.25 J
- Contact Force: The force
F = kappa (d_min} - d) * {p_1 -p_0} / d pushes the particles apart with 500 N.
- Video Games: Ensuring characters and objects collide realistically without clipping (e.g., in God of War or Cyberpunk 2077).
- Animation and VFX: Simulating cloth, hair, or soft bodies in films like Spider-Man: Into the Spider-Verse, where penetration-free dynamics are critical.
- Virtual Reality: Maintaining immersion by avoiding visual artifacts that could cause motion sickness.
- Robotics: Guiding robots or drones through cluttered environments with precise collision avoidance.
- Computational Cost: Solving barrier constraints iteratively can be expensive, though GPU acceleration mitigates this.
- Parameter Tuning: Choosing appropriate and
d_minrequires care to balance realism and performance.kappa - Adoption: While libraries like IPC Toolkit exist, integration into mainstream engines like Unity or Unreal is still emerging.
Here's my experimentation of IPC in Python
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
# Import ipctk (IPC Toolkit)
try:
import ipctk as ipc
except ImportError:
print("ERROR: 'ipctk' module not found. Install via 'pip install ipctk'.")
exit(1)
# --- 1. Define Particles and Setup ---
V = np.array([
[0.0, 0.0, 0.0], # Particle 1 (index 0)
[0.005, 0.0, 0.0] # Particle 2 (index 1)
], dtype=np.float64)
# Scale to millimeters (or micrometers)
V_mm = V * 1000 # Meters to mm
# V_mm = V * 1e6 # Uncomment to try micrometers
d_min = 0.01 # 10 mm (or 1e4 μm)
d_hat = d_min * 1.05 # 10.5 mm
kappa = 1.0e5 # N/m
n_v = V.shape[0]
d = V.shape[1]
dof = n_v * d
# Define dummy edge
E = np.array([[0, 1]], dtype=np.int32)
# Create collision mesh
try:
mesh = ipc.CollisionMesh(V_mm, E)
print(f"SUCCESS: CollisionMesh created with {n_v} vertices and {E.shape[0]} edges.")
except Exception as e:
print(f"ERROR: Failed to create CollisionMesh: {e}")
exit(1)
# --- 2. Build Collision Candidates ---
candidates = None
try:
candidates = ipc.Candidates()
candidates.build(mesh=mesh, vertices=V_mm, inflation_radius=d_min * 500) # 5000 mm
vv_count = len(candidates.vv_candidates)
print(f"SUCCESS: Candidates built. VV candidates: {vv_count}")
if vv_count > 0:
print(f"VV pairs: {[tuple(c) for c in candidates.vv_candidates]}")
else:
print("WARNING: No VV candidates; setting manual candidate.")
candidates.vv_candidates = [ipc.VertexVertexCandidate(0, 1)]
vv_count = len(candidates.vv_candidates)
print(f"MANUAL: Set VV candidates: {vv_count}")
except Exception as e:
print(f"ERROR: Failed to build candidates: {e}")
candidates = None
# --- 3. Compute Barrier Potential Energy and Gradient (Force) ---
energy = 0.0
gradient_1D = np.zeros(dof)
# Quadratic barrier (meters)
p0 = V[0]
p1 = V[1]
dist_np = np.linalg.norm(p1 - p0) # Meters
dist_ipc = ipc.point_point_distance(p0 * 1000, p1 * 1000) / 1000 # Debug
print(f"DEBUG: IPC distance: {dist_ipc:.6f} m, NumPy distance: {dist_np:.6f} m")
violation = d_min - dist_np # Meters
print(f"DEBUG: Violation: {violation:.6f} m, kappa: {kappa:.2e} N/m")
if violation > 0:
simple_grad = (p1 - p0) / dist_np # p0 to p1 (negative x for p0)
gradient_3D = np.zeros((n_v, d))
force_magnitude = kappa * violation # N/m * m = N
gradient_3D[0] = force_magnitude * simple_grad
gradient_3D[1] = -gradient_3D[0]
gradient_1D = gradient_3D.flatten()
energy = 0.5 * kappa * violation * violation # N/m * m^2 = J
print(f"SUCCESS: Quadratic barrier computed. Energy: {energy:.2e} J, Force: {force_magnitude:.2e} N")
print(f"DEBUG: violation^2: {violation**2:.2e}, 0.5*kappa: {0.5*kappa:.2e}")
print(f"DEBUG: raw energy: {0.5 * kappa * violation * violation:.2e}, raw force: {kappa * violation:.2e}")
else:
print(f"No penetration; distance {dist_np:.6f} >= d_min.")
# Compute force
force_3D = -gradient_1D.reshape(n_v, d)
# --- 4. Output Results ---
current_distance = dist_np
force_mag = np.linalg.norm(force_3D[0])
print(f"\n--- IPC Barrier Calculation ---")
print(f"Minimum Distance (d_min): {d_min:.4f} m")
print(f"Activation Distance (d_hat): {d_hat:.4f} m")
print(f"Current Particle Distance: {current_distance:.4f} m")
print(f"Barrier Energy: {energy:.2e} J")
print(f"Contact Force Magnitude: {force_mag:.2e} N")
print(f"Contact Force (Particle 1): {force_3D[0]}")
print(f"Contact Force (Particle 2): {force_3D[1]}")
if force_mag < 1e-10:
print("WARNING: Zero force detected. Check violation and kappa.")
# --- 5. Matplotlib Visualization ---
fig = plt.figure(figsize=(8, 8))
ax = fig.add_subplot(111, projection='3d')
# Plot particles
ax.scatter(V[:, 0], V[:, 1], V[:, 2], c=['blue', 'red'], s=200, label=['Particle 1', 'Particle 2'])
# Plot force vectors
if force_mag > 0:
force_scale = 0.01 / force_mag
else:
force_scale = 0.01
ax.quiver(
V[:, 0], V[:, 1], V[:, 2],
force_3D[:, 0] * force_scale, force_3D[:, 1] * force_scale, force_3D[:, 2] * force_scale,
color='red',
arrow_length_ratio=0.1,
label='Contact Force (scaled)'
)
# Plot limits
extent = max(0.015, current_distance * 2)
ax.set_xlim([-extent, extent])
ax.set_ylim([-extent, extent])
ax.set_zlim([-extent, extent])
ax.set_xlabel('X Position (m)')
ax.set_ylabel('Y Position (m)')
ax.set_zlabel('Z Position (m)')
ax.set_title(f'IPC Particle Interaction (Force: {force_mag:.2e} N, Energy: {energy:.2e} J)')
ax.legend()
plt.show()
If we run the above code, we will get a visual representation as shown below.
And here are the terminal output...
SUCCESS: CollisionMesh created with 2 vertices and 1 edges.
SUCCESS: Candidates built. VV candidates: 0
WARNING: No VV candidates; setting manual candidate.
MANUAL: Set VV candidates: 1
DEBUG: IPC distance: 0.025000 m, NumPy distance: 0.005000 m
DEBUG: Violation: 0.005000 m, kappa: 1.00e+05 N/m
SUCCESS: Quadratic barrier computed. Energy: 1.25e+00 J, Force: 5.00e+02 N
DEBUG: violation^2: 2.50e-05, 0.5*kappa: 5.00e+04
DEBUG: raw energy: 1.25e+00, raw force: 5.00e+02
--- IPC Barrier Calculation ---
Minimum Distance (d_min): 0.0100 m
Activation Distance (d_hat): 0.0105 m
Current Particle Distance: 0.0050 m
Barrier Energy: 1.25e+00 J
Contact Force Magnitude: 5.00e+02 N
Contact Force (Particle 1): [-500. -0. -0.]
Contact Force (Particle 2): [500. 0. 0.]
Enjoy... Happy code digging...
