Analysis of the particle relaxation method for generating uniform particle distributions in smoothed particle hydrodynamics

Understand

We establish a theoretical framework of the particle relaxation method for uniform particle generation of Smoothed Particle Hydrodynamics.

• We achieve this by reformulating the particle relaxation as an optimization problem.
• The objective function is an integral difference between discrete particle-based and smoothed-analytical volume fractions.
• The analysis demonstrates that the particle relaxation method in the domain interior is essentially equivalent to employing a gradient descent approach to solve this optimization problem, and we can extend such an equivalence to the bounded domain by introducing a proper boundary term.