Extension of the fast multipole method for the rectangular cells with an anisotropic partition tree structure

Area of Science:

• Computational Physics
• Materials Science
• Chemistry

Background:

• The Fast Multipole Method (FMM) is crucial for efficient electrostatic interaction calculations in molecular dynamics (MD).
• Conventional FMM implementations are limited to cubic unit cells with isotropic periodic boundary conditions.
• This limitation restricts MD simulations to specific system geometries.

Purpose of the Study:

• To extend the FMM algorithm for adaptive calculations in rectangular MD unit cells with anisotropic periodicity.
• To enable more flexible and general MD simulations by removing geometric constraints.
• To maintain the numerical rigor and accuracy of electrostatic interaction calculations.

Main Methods:

• Developed an anisotropic hierarchical partitioning scheme for the FMM.
• Implemented the extended FMM into parallelized general-purpose MD software.
• Utilized a partition tree that can include binary and ternary branches, chosen arbitrarily along coordinate axes.
• Analyzed errors associated with the partition tree structure.

Main Results:

• Successfully adapted the FMM for rectangular unit cells with varying periodicity along different axes.
• The extended FMM maintains high accuracy comparable to the conventional method.
• The algorithm is suitable for systems with uniformly distributed point charges.
• The partition tree flexibility allows for arbitrary branching structures.

Conclusions:

• The extended FMM significantly broadens the applicability of MD simulations to systems with diverse unit cell shapes and boundary conditions.
• This advancement allows for more general and accurate electrostatic calculations in computational studies.
• Future extensions to other prime number branches (e.g., 5, 7) are feasible.