AI 中文总结
本文提出三种基于SPH(2)的粒子方法，通过广义垂直坐标变换提升复杂底部边界自由表面流模拟效率，首次将σ坐标系应用于粒子法，验证了方法的有效性。
AI 中文摘要
针对具有复杂底部边界的自由表面流问题，我们提出三种新的粒子方法，通过引入广义垂直坐标变换（VCT）提升计算效率。第一种是底部边界拟合粒子方法（BF-SPH），它将有限差分法中的贴体坐标系简单应用于粒子法，通过将复杂底部变换为平坦底部，可准确施加底部边界条件且操作简便。第二种是底部边界拟合椭球粒子方法（BFE-SPH），它将BF-SPH与Shibata等人提出的椭球粒子模型结合，通过选择合理的椭球粒子纵横比加快粒子模拟速度。第三种是σ-SPH方法，它利用σ坐标系根据水深自动选择椭球粒子的纵横比，σ坐标系常用于海洋场数值模拟，如普林斯顿海洋模型，这是首次将其应用于粒子法的尝试。海洋问题中，如海啸的3-D粒子法详细分析需从近海到沿海区域的垂直分辨率，σ坐标系可通过参考水深逐步过渡到自然高效的坐标系。本文证明上述三种方法可统一为垂直坐标变换（VCT），且通过采用具有二阶导数（含交叉导数）二阶精度的SPH(2)成功实现VCT。
英文摘要
We propose three new particle methods that improve computational efficiency by introducing a generalized Vertical Coordinate Transformation (VCT) for free surface flow problems with complex bottom boundaries. The first method is a bottom boundary-fitted particle method (BF-SPH). The BF-SPH is simply an arrangement of the body-fitted-coordinate system in the finite difference method to the particle method. The BF-SPH can accurately impose the bottom boundary conditions, while a simple procedure is performed by transforming the complex bottom into a flat one. The second method is the bottom boundary-fitted ellipsoidal particle method (BFE-SPH), which combines the BF-SPH with the ellipsoidal particle model proposed by Shibata et al. The BFE-SPH can speed up the particle simulation by choosing a reasonable aspect ratio of ellipsoidal particles. The last method is the $σ$-SPH method, which automatically selects the aspect ratios of ellipsoidal particles concerning water depth using the $σ$-coordinate system. The $σ$-coordinate is often employed in numerical simulations of oceanographic fields, such as in the Princeton Ocean Model. However, this is the first attempt to apply the $σ$-coordinate to a particle method. Vertical resolution is required from offshore to the coastal region in oceanographic problems such as tsunamis, especially when conducting detailed analysis using a 3-D particle method. Using the $σ$-coordinate allows for a stepwise transition to a naturally efficient coordinate system by referencing water depth. In this paper, we have shown that the above three methods can be generalized as Vertical Coordinate Transformations (VCTs), and the VCTs are successfully achieved by employing SPH(2) with the second-order accuracy of the second-order derivatives, including cross derivatives.
Journal refJournal of Computational Physics 543 (2025) 114407
