Jello Cube Simulation

Simulating a Mass-Spring System
Pranav Rathod

Date: February 19, 2025

For this assignment, I successfully implemented a physically based simulation of a Jello Cube using a mass-spring system. The cube deforms, oscillates, and reacts realistically to forces based on Newton’s laws of motion, Hooke’s law, and damping forces.

Video Demonstration

Core Features Implemented:

• Acceleration Computation (computeAcceleration() in physics.cpp):

  • Structural, shear, and bend spring forces
  • External force field (if present)
  • Collision response with the bounding box

  The elasticity of the Jello Cube is modeled using a mass-spring system, where forces follow Hooke’s Law:

  $$F = -k \cdot (x - x_0)$$
  • F: restoring force
  • k: spring constant
  • x: current length
  • x0: rest length

  Each mass point connects to others via structural, shear, and bend springs, allowing the cube to deform and recover naturally.

  • Damping Forces: To stabilize the system and reduce oscillations, damping forces are computed as:

    $$F_d = -k_d \cdot \frac{(v_A - v_B) \cdot L}{|L|}$$
    • k_d: damping coefficient
    • (v_A - v_B): velocity difference
    • L: displacement vector between connected points

• Euler and RK4 Integration: Both integration methods were implemented to advance the simulation over time with tunable timestep values.

  • Collision Detection & Penalty Response (Bounding Box): Ensures all control points stay within [-2, 2] by applying a spring force when out of bounds.

    $$F_c = k_c \cdot (p_{boundary} - p)$$

• External Force Field: Trilinear interpolation was used to calculate external force values from a 3D grid defined in the .w file.

  1. Grid Indexing – mapping position to force grid
  2. Interpolation – blending 8 surrounding vectors

• OpenGL Visualization:

  • Wireframe and shaded surface rendering modes
  • Real-time camera controls
  • Toggle visibility for structural, shear, and bend springs

Extra Credit Implemented

Inclined Plane Collision

• Supported via user-defined coefficients in the world file:

  $$a \cdot x + b \cdot y + c \cdot z + d = 0$$

• Rendering: Drawn dynamically in showInclinedPlane() using 4 corners based on the plane equation.

• Collision Detection & Response:

  • Each control point is checked against the plane.
  • If below, a penalty force pushes it above the surface.
  • Includes damping opposite to velocity to absorb energy.

• Implemented in applyCollisionForces() inside physics.cpp.

The inclined plane adds more complexity and realism to the cube’s motion and response.

Development Environment

• OS: MacOS
• Editor: Visual Studio Code
• Compiler: Clang++ (C++17)
• Build System: Makefile
• Libraries: OpenGL, GLUT (Mac Frameworks)

Compilation Instructions

1. Build the project

   make

2. Run the simulation with a world file

   ./jello world/jello.w

3. Create a world file

   ./createWorld

4. Clean build artifacts

   make clean

Note: The Makefile auto-detects MacOS and links the correct OpenGL/GLUT frameworks.

Notes

• The simulation was tested using multiple world files including custom-generated ones.
• Performance is interactive with frame rates > 15fps at 640x480.
• A Mac executable is included, but the grader may recompile on Windows as needed.