AI 中文总结
本文提出一种保结构ALE-BGN-MDR方法，解决含移动接触线与重力的Navier-Stokes自由边界问题，消除重力离散的一致性误差，通过数值实验验证了方法的理论性质。
AI 中文摘要
针对含移动接触线的不可压缩Navier-Stokes自由边界问题，本文提出一种与重力一致的任意拉格朗日-欧拉（ALE）有限元方法。重力的直接体力离散可能无法保证离散重力功与演化域上重力势能变分的一致性，从而产生人为一致性误差并在平衡态附近持续出现虚假速度。为消除该不一致性，本文将重力势能变分重新表述为中间ALE构型上的移动边界积分，并采用辛普森求积规则对其进行精确计算，由此得到一种弱非线性全离散格式，其中重力贡献与离散势能变分完全一致。该方法可精确保持体积，满足包含重力势能的离散能量耗散定律，且在适当假设下可使离散速度在长时间尺度上趋于零，从而排除持续的重力诱导虚假速度。结合自由表面的BGN处理与体网格扩展的MDR方法，该格式可在移动接触线附近维持精确的界面追踪与良好的网格质量。二维与三维空间的数值实验验证了上述理论性质。
英文摘要
We propose a gravity-consistent arbitrary Lagrangian--Eulerian finite element method for incompressible Navier--Stokes free-boundary problems with moving contact lines. A direct body-force discretization of gravity may fail to ensure consistency between the discrete gravitational work and the variation of the gravitational potential energy on the evolving domain, resulting in an artificial consistency error and persistent spurious velocities near equilibrium. To remove this inconsistency, we reformulate the gravitational potential energy variation as a moving-boundary integral over intermediate ALE configurations and evaluate it exactly using Simpson's quadrature rule. This leads to a mildly nonlinear fully discrete scheme in which the gravitational contribution is exactly consistent with the discrete potential-energy variation. The proposed method preserves volume exactly, satisfies a discrete energy-dissipation law including gravitational potential energy, and under suitable assumptions, drives the discrete velocity to zero in the long-time regime, thereby excluding persistent gravity-induced spurious velocities. Together with the BGN treatment of the free surface and the MDR bulk mesh extension, the scheme maintains accurate interface tracking and good mesh quality near the moving contact line. Numerical experiments in two and three spatial dimensions confirm the theoretical properties.
