Question on Handling Distance Gradient Discontinuities

[coreqode]

Hello,

I'm looking into the details of the Incremental Potential Contact (IPC) method and have a question regarding the continuity of the distance function's derivatives. I would appreciate any clarification you could provide.

My understanding is that IPC relies on the exact distance $d$ between pairs of codimensional primitives (e.g., point-triangle, edge-edge).
However, the global distance function is defined as the minimum over all possible pairs. This piecewise definition makes its gradient discontinuous. For instance, as a vertex moves, its closest point on the opposing mesh might transition from the interior of a triangle to one of its edges. At this transition boundary, the analytical form of the distance function changes, leading to a jump in its gradient ($\nabla d$) and Hessian ($\nabla^2 d$).

My question is: How does the IPC algorithm maintain robustness and convergence for its Newton-based solver despite these discontinuities in the distance function's derivatives?

[zfergus]

This is a great question. There are two parts to the continuity of the distance functions.

Sub-Element Continuity

The first possible source of discontinuity is when a collision pair changes the closest sub-element. For example, when a point is closest to a triangle's interior, but then goes off the edge, and the closest point is on the edge. Similarly, the closest point can be a vertex of the triangle.

In these instances, the function is $C^\infty$ on the interior and $C^1$ at the subelement transition[1]. This Hessian discontinuity is somewhat rare and doesn't seem to affect the optimization too negatively.

Global Continuity

You are right that if we did a global minimum of distance to all elements, this would be discontinuous at the transition.

Instead, we simply sum up the barrier energies for all elements within the $\hat{d}$ distance,

$$ B(x) = \kappa \sum_i b(d_i(x), \hat{d}). $$

The obvious problem with this approach is that it introduces duplicate contact forces at transition points. This problem has been explored by several follow-up works. For instance, in Convergent IPC, we subtract away certain sub-element terms to remove the duplication.

Here is a short list of papers that address this issue:

1. Convergent Incremental Potential Contact
2. Robust and Artefact-Free Deformable Contact with Smooth Surface Representations
3. Offset Geometric Contact
4. Geometric Contact Potential

Of these, 1 is already implemented in the toolkit, and 3 and 4 are current PRs.

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[1] The only instance where the transition is only $C^0$ is when edges are parallel, as the line-line distance function degenerates at that point. For this reason, we mollify the distance function based on the angle. Please refer to the original paper for more details.