On the Hessian of Shape Matching Energy

TL;DR

This paper derives the analytic Hessian matrix for shape matching energy, enabling its stable use with implicit integrators in meshless deformation simulations.

Contribution

It provides the first explicit derivation of the Hessian for shape matching energy, facilitating its application in implicit integration methods.

Findings

Derived the analytic Hessian matrix for shape matching energy.

Enables stable implicit integration in shape matching simulations.

Facilitates broader application of shape matching in real-time dynamics.

Abstract

In this technical report we derive the analytic form of the Hessian matrix for shape matching energy. Shape matching is a useful technique for meshless deformation, which can be easily combined with multiple techniques in real-time dynamics. Nevertheless, it has been rarely applied in scenarios where implicit integrators are required, and hence strong viscous damping effect, though popular in simulation systems nowadays, is forbidden for shape matching. The reason lies in the difficulty to derive the Hessian matrix of the shape matching energy. Computing the Hessian matrix correctly, and stably, is the key to more broadly application of shape matching in implicitly-integrated systems.