How does torch-diffsim work?

This section provides a description of how torch-diffsim implements differentiable physics simulation. We cover both the forward simulation (physics) and the backward differentiation (gradients).

Overview

torch-diffsim combines two key components:

1. Physics Simulation: A semi-implicit (symplectic Euler) time integrator for tetrahedral finite element method (FEM) using the Stable Neo-Hookean hyperelastic material model

2. Automatic Differentiation: Gradient computation through the simulation using PyTorch’s autograd, enabling gradient-based optimization of material properties, initial conditions, and other parameters

• Simulation
  • Finite Element Method (FEM)
  • Stable Neo-Hookean Material
  • Semi-Implicit (Symplectic Euler) Time Integration
  • Contact Handling
  • Mass Matrix
  • Boundary Conditions
  • Implementation Notes
• Differentiation
  • Automatic Differentiation Overview
  • Differentiable Simulation
  • Gradient Flow Through Time Steps
  • Energy-Based Force Computation
  • Material Parameter Gradients
  • Differentiable Contact
  • Smooth Operations for Differentiation
  • Memory-Efficient Backpropagation
  • Spatially Varying Materials
  • Optimization Example
  • Practical Considerations
  • Verification

Core Principles

• Symplectic Integration

The simulator uses a first-order symplectic method (semi-implicit Euler) that provides better energy conservation than explicit methods.

• Smooth Operations

All operations are smooth (differentiable) - no hard constraints, projections, or conditional branching that would break gradient flow.

• Energy-Based Formulation

Forces are derived as negative gradients of the total elastic energy, which naturally integrates with PyTorch’s autograd system.

• Barrier Functions

Contact handling uses smooth barrier potentials instead of hard constraints, maintaining \(C^2\) continuity for gradient computation.

Mathematical Notation

Throughout this documentation, we use the following notation:

Symbol	Meaning
\(\mathbf{x}^n\)	Vertex positions at time step \(n\), \((N \times 3)\) tensor
\(\mathbf{v}^n\)	Vertex velocities at time step \(n\), \((N \times 3)\) tensor
\(\mathbf{M}\)	Mass matrix (diagonal), \((N \times N)\) matrix
\(\mathbf{f}\)	Force vector, \((N \times 3)\) tensor
\(\mathbf{F}\)	Deformation gradient, \((M \times 3 \times 3)\) tensor
\(\Psi(\mathbf{F})\)	Strain energy density function
\(\Delta t\)	Time step size
\(N\)	Number of vertices
\(M\)	Number of tetrahedral elements
\(E\)	Young’s modulus (material parameter)
\(\nu\)	Poisson’s ratio (material parameter)