  Brownian dynamics of colloidal particles on complex surfaces has found
important applications in diverse physical, chemical and biological processes.
However, current Brownian dynamics simulation algorithms mostly work for
relatively simple surfaces that can be analytically parameterized. In this
work, we develop an algorithm to enable Brownian dynamics simulation on
extremely complex surfaces. We approximate complex surfaces with triangle mesh
surfaces and employ a novel scheme to perform particle simulation on these
triangle mesh surfaces. Our algorithm computes forces and velocities of
particles in global coordinates but updates their positions in local
coordinates, which benefits from the advantages of simulation schemes in both
global and local coordinate alone. We benchmark the proposed algorithm with
theory and then simulate Brownian dynamics of both single and multiple
particles on torus and knot surfaces. The results show that our method captures
well diffusion, transport, and crystallization of colloidal particles on
complex surfaces with non-trivial topology. This study offers an efficient
strategy for elucidating the impact of curvature, geometry, and topology on
particle dynamics and microstructure formation in complex environments.
