  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. 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. This insight enables us to
control the relaxed particle distribution by selecting the target kernel
cut-off radius for a given kernel function.
