XPBD: Position-Based Simulation of Compliant Constrained Dynamics

How to read this

Category

Method: a constraint-dynamics formulation extending Position-Based Dynamics

Contributions

• XPBD, an extension of PBD that removes the dependence of constraint stiffness on time step and iteration count
• Derives from an implicit position-level discretization and introduces a total Lagrange multiplier giving constraints a correspondence to elastic and dissipative energy potentials, plus accurate constraint-force estimates
• Solves constraints at the position level (Gauss-Seidel or Jacobi) via a compliance matrix (inverse stiffness) with a Rayleigh dissipation damping term

Context

Directly extends Position Based Dynamics (Muller et al.) and the Unified Particle Physics framework (Macklin et al.), grounding stiffness in physically meaningful compliance.Builds on: Position Based Dynamics · Unified Particle Physics for Real-Time Applications

Correctness

Validated against a reference non-linear Newton solver on harmonic oscillators, hanging chains, cantilever beams, cloth, and inflatable balloons, showing visually indistinguishable results with one extra scalar per constraint; it remains a real-time-oriented approximation, not a guaranteed-accurate Newton solve.

Clarity

Clear and well-motivated; a first pass conveys why and how it fixes PBD stiffness, with the derivation rewarding a second pass.

How to read it

First pass for the stiffness-independence problem and the compliance/total-multiplier fix; second pass on the implicit derivation and per-constraint update if you implement it or need force estimates.