Analysis of the particle relaxation method

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, which globally measures zero-order inconsistency in the interior domain. 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. Additionally, each periodic particle distribution has a spatially uniform particle volume, denoted as characteristic volume. The relaxed particle distribution has the largest characteristic volume, and the kernel cut-off radius determines this volume. These insights explain the equivalence between uniform particle distributions and optimized zero-order consistency and enable us to control the relaxed particle distribution by selecting the target kernel cut-off radius for a given kernel function.