An older post (https://csmbrannon.net/2013/09/29/particle-basis-function-in-the-cpdi-method/) showed particle basis functions in 1D, demonstrating that they retain overlapping support with neighboring particles when using CPDI even when the particles are stretched to be large multiples of the grid cell length. In the 1D context of those examples, “holes” in CPDI basis functions are impossible.

This post shows an extreme 2D example (suggested by John Nairn, Jim Guilkey, and Michael Homel*) that can deleteriously lose overlapping support of particle shape functions, which can then allow non-physical material interpenetration.

Once the interpenetration begins, though, I had expected some degree of overlapping support to develop. I had expected a “dip” to form in the particle basis function for the middle-top particle in both legs of the T (bringing down the basis functions for each to, say, 1/2, so that they are both nonzero and form a partition of unity). Instead, my code is predicting that the very top particle gets eaten away so its value goes to 0 while the other particle continues to have a shape function value of 1.   This behavior does indeed correspond to formation of a hole in the particle basis function.  Convergence is not destroyed since mesh refinement should ultimately restore overlapping particle basis function support, but this does point at a need for as-yet-undiscovered means of monitoring run-time solution quality to prevent material interpenetration with fractional particles. Take a look at the pictures below, and then a summary follows.

Recall that Ψ_pi was defined in the older post to equal Φ_ip divided by the sum over all p of Φ_ip. Because of this normalization, my intuition was that, once spurious material interpenetration began, there would be an even averaging of overlapping particle basis functions that would produce a dip in both, not slice through one of them as seen here.  If we had gotten the expected dip, it would have provided at least some overlapping support to resist interpenetration (like an automatic contact algorithm). Alas, such is not the case, and we do not yet know what algorithm changes are needed to allow this coarse discretization case-study problem to run without interpenetration.  Got any ideas?  One simple (as-yet untested) idea is to subdivide the particle into subdomains of size smaller than grid spacing. This idea, which is analogous to subcycling in time integrators (so I will give it a similar name: sub-sampling), is quite different from particle splitting — no new particles would be created with this approach, and subsampling would be invoked only when particles are significantly stretched, so it would not be needed when a particle recompresses, thus making it far more attractive than particle splitting.  While this approach would probably avoid material interpenetration, one should be cautioned that it might present accuracy issues. The advantage of true particle splitting is that you can then allow the newly created particles to accumulate different levels of strain, which wouldn’t be possible with sub-sampling (which would continue to presume that the deformation is affine over the entire particle, whether sub-sampled or not). Thus, in regions where accuracy of the predication is important, then particle splitting would be recommended. In regions where errors would be acceptable (such as far away from an explosion where your only goal is to “catch” debris without accurate predictions of the stresses induced in the catching surface), sub-sampling would be at great choice.

Shortly after this post, Professor John Nairn (Oregon State) provided the following demonstration simulations of both proper non-interpenetration and improper pure pass through of the T-bar (dart) problem.

Non-penetration: the middle particle domain is short enough to ensure that its particle basis function has overlapping support with that of the dart…

Pure pass through: if the middle particle is stretched vertically by a significantly larger amount than the non-interpenetration case shown above, then the middle particle will continue to interact with its neighbors above and below, but it will not have overlapping support in the horizontal direction with relatively skinny particles (like the tip of the dart).

Besides complete transfer of momentum in first plot (i.e., the horizontal dart stops), Professor Nairn reports that he can get partial transfer and partial pass though by varying when the dart reaches the bar. This makes sense. If a particle that is near (but not beyond) the length threshold for interpenetration, then the stretching from impact will carry it beyond the length for interpenetration.

And now some e-self-flagellation :

THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!
THERE CAN BE HOLES IN PARTICLE SHAPE FUNCTIONS!

FOOTNOTE
* John Nairn and Jim Guilkey provided the specific case study (T-bar/dart) demonstrating that holey particle shape functions can indeed occur, and Michael Homel provided an insightful sketch of why he believed that their case study would indeed all material interpenetration.