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: Tetrahedral Stable Neo-Hookean FEM with either a semi-implicit (symplectic Euler) step or an optional backward-Euler IPC solve

2. Differentiation: Unrolled PyTorch autograd for the explicit solver and an implicit adjoint for each converged IPC step

• 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

• Semi-Implicit Integration

The explicit solver uses a first-order semi-implicit Euler update. That update is symplectic before the solver’s damping, velocity limiting, constraints, and contact response are applied.

• Smooth Operations

The differentiable solver uses unrolled PyTorch operations. Gradients are local and piecewise around contact activation.

• Energy-Based Formulation

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

• Barrier Functions

Differentiable ground contact uses a compact \(C^2\) penalty instead of hard projection. The IPC solver instead uses the IPC barrier together with conservative CCD.

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)