Physics-Based Simulation

Summary

In this lecture, we discussed modal reduction techniques that dramatically accelerate physics-based simulations by projecting high-dimensional systems onto carefully chosen low-dimensional subspaces.

Linear modal analysis uses an object's natural vibration modes as basis vectors, creating independent 1D oscillators that are extremely efficient to solve. However, this approach produces severe artifacts (length stretching, volume changes) for large deformations due to its linear assumptions. Nonlinear model reduction extends these concepts to handle large deformations by projecting the full nonlinear equations of motion. The key challenge is efficiently evaluating reduced forces without computing full-space quantities, solved through polynomial fitting for specific materials or cubature methods for general cases.

The quality of any modal reduction depends critically on basis selection. Two main approaches exist: data-driven methods using Principal Component Analysis on simulation snapshots, and physics-based modal derivatives that capture nonlinear coupling between linear modes without requiring expensive pre-simulations.