Section: New Results

Hierarchical Adaptively Restrained Particle Simulations

Participants : Svetlana Artemova, Stephane Redon.

It has been shown that algorithms relying on hierarchical representations of molecular systems may accelerate molecular simulations: for example, divide-and-conquer approach for simulations in internal coordinates [10] , [11] , adaptive algorithms for dynamics of articulated bodies [15] , algorithms for neighbor search for system with symmetries [12] or for large rigid molecules [8] .

Therefore, we were interested in combining hierarchically-based algorithms with Adaptively Restrained Particle Simulations (ARPS). Precisely, as with classical ARPS, we have considered the adaptively restrained (AR) Hamiltonian:

but we have introduced a different form of the inverse inertia matrix $\Phi (𝐪,𝐩)$. In this case, again, positional degrees of freedom are adaptively switched on and off during the simulation, but, these are relative positional degrees of freedom in the system, and not the positional degrees of freedom of individual particles. Precisely, particles are grouped together into rigid bodies according to the tree representation and released repeatedly during the simulation. We call this approach hierarchical Adaptively Restrained Particle Simulations (hierarchical ARPS).

We have performed several numerical experiments to illustrate this new approach. For example, in Fig. 8 we present the planar collision cascade study.

For hierarchical AR simulations, obtained results depend on the tree representation of the system: for the results demonstrated in Fig. 8 the tree was constructed in a top-down manner by recursive dividing of the system in halves and, therefore, the squares of different levels are being activated by the shock.

Figure 8. Simulating a collision cascade with controlled precision. Hierarchical adaptively restrained simulations allow us to smoothly trade between precision and speed. The main features of the shock are preserved. The binary tree representation was constructed top-down.

To clearly demonstrate the effect of the tree, we provide the results for the same four simulations with another tree built in a bottom-up manner by grouping the particles pairwise according to their sequence number (they were enumerated, first, along the $y$-axis, vertically, and then, along the $x$-axis, horizontally). These results are shown in Fig. 9 , and are rather different from those in Fig. 8 : vertical lines are being activated when the central part of the plane is reached by the shock.

Figure 9. Simulating a collision cascade with controlled precision. Hierarchical adaptively restrained simulations allow us to smoothly trade between precision and speed. The main features of the shock are preserved. The binary tree representation was constructed bottom-up.

The patent reporting the principles and the algorithms used to implement hierarchical ARPS has been deposited.