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VBD’s per-vertex 3×3 solve already gives the locally optimal descent direction — there is no leverage left inside a single block. So why does it still slow down on problems with high stiffness contrast? The answer turns out to be a story about what basis you descend in, and once that picture is in place the fix — a sparse coarse correction built from per-element eigenmodes — almost suggests itself.
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A practical primer covering exactly the quaternion operations used in rigid body simulation, with reference to the Newton AVBD implementation. No proofs, just what you need to read the code.
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This post derives the per-vertex 3×3 Hessian block for the stable Neo-Hookean tet material under VBD-style block Gauss-Seidel, and shows how it lands as an unconditionally PSD expression with no clamp or eigenvalue projection required. The derivation is short but the algebraic cancellation it relies on is easy to miss, so it is worth writing out in full. The post is meant as a reference for anyone wiring stable Neo-Hookean into a VBD solver.
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In the VBD paper (SIGGRAPH 2024), we briefly discuss extending Vertex Block Descent to rigid body simulation. The idea is natural: instead of updating a single vertex with 3 DoF, you update an entire rigid body with 6 DoF. But the details matter. This post walks through the full derivation—from the continuous Newton-Euler equations, to discrete backward Euler as a nonlinear system, to the Schur complement solve you actually run each iteration—with reference code from Newton, which implements this approach under the name AVBD (Augmented VBD).
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Vertex Block Descent (VBD) is a physics solver we published at SIGGRAPH 2024 for elastic body dynamics. It offers unconditional stability, excellent GPU parallelism, and fast convergence to implicit Euler solutions. While the paper covers the formulation comprehensively, actually implementing VBD correctly—especially the damping—turns out to be subtler than it first appears. This post discusses the key pitfalls and how to get them right, based on lessons learned during development with NVIDIA Warp.
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