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.

翻译：我们建立了用于光滑粒子流体动力学均匀粒子生成的粒子松弛法的理论框架。通过将粒子松弛问题重新表述为优化问题来实现这一目标。目标函数为离散粒子基体积分数与光滑解析体积分数之间的积分差异。分析表明，区域内部的粒子松弛法本质上等价于采用梯度下降法求解该优化问题，且通过引入合适的边界项可将这种等价性推广到有界区域。此外，每个周期性粒子分布均具有空间均匀的粒子体积，称为特征体积。松弛后的粒子分布具有最大的特征体积，该体积由核函数截断半径决定。这一认识使得我们能够通过为给定核函数选择目标核函数截断半径来控制松弛后的粒子分布。