This paper introduces a boundary condition scheme for weakly compressible (WC) renormalised first-order accurate meshless Lagrangian methods (MLM) by considering both solid and free surface conditions.

A hybrid meshless Lagrangian method-finite difference (MLM-FD) scheme on prescribed boundary nodes is proposed to enforce Neumann boundary conditions. This is used to enforce symmetry boundary conditions and the implied Neumann pressure boundary conditions on solid boundaries in a manner consistent with the Navier-Stokes equation leading to the accurate recovery of surface pressures. The free surface boundary conditions allow all differential operators to be approximated by the same renormalised scheme while also efficiently determining free surface particles.

The boundary conditions schemes are implemented for two renormalised MLMs. A WC smoothed particle hydrodynamics (SPH) solver is compared to a WC generalised finite difference (GFD) solver. Applications in both 2D and 3D are explored. A substantial performance benefit was found when comparing the WCGFD solver to the WCSPH solver with the WCGFD solver realising a maximum speedup in the range of three times over WCSPH in both 2D and 3D configurations. The solvers were implemented in C++ and used the NVIDIA CUDA 10.1 toolkit for the parallelisation of the solvers.

本文通过考虑固体和自由表面条件，介绍了弱可压缩 (WC) 重归一化一阶精确无网格拉格朗日方法 (MLM) 的边界条件方案。

提出了一种在指定边界节点上的混合无网格拉格朗日法-有限差分 (MLM-FD) 方案来强制执行 Neumann 边界条件。这用于以与 Navier-Stokes 方程一致的方式在固体边界上强制实施对称边界条件和隐含的 Neumann 压力边界条件，从而导致表面压力的准确恢复。自由表面边界条件允许所有微分算子通过相同的重整化方案来近似，同时还可以有效地确定自由表面粒子。

边界条件方案是为两个重新归一化的 MLM 实现的。WC 平滑粒子流体动力学 (SPH) 求解器与 WC 广义有限差分 (GFD) 求解器进行了比较。探索了 2D 和 3D 中的应用。将 WCGFD 求解器与 WCSPH 求解器进行比较，发现 WCGFD 求解器与 WCGFD 求解器在 2D 和 3D 配置中实现的最大加速比 WCSPH 高三倍时，发现了显着的性能优势。求解器以 C++ 实现，并使用 NVIDIA CUDA 10.1 工具包进行求解器的并行化。