自由液面流动问题在航天航空工程，船舶海洋工程和水利工程中有着广泛的应用背景。此问题涉及非线性的流体动力学模型以及自由液面的飞溅、融合、破碎、翻卷等拓扑非线性变化。基于网格的数值方法在流动大变形、自由液面追踪和流固耦合方面存在着很大的困难和挑战，而采用人工状态方程的弱可压物质点法存在效率低下、压力振荡严重和难于施加压力边界条件的问题。为了克服这些缺点，本文针对自由液面流动问题提出了一种完全不可压物质点法。新算法采用算子分裂思想解耦了流体N-S方程的压力项和速度项，大大提高了显式计算的时间步长，并通过全场的压力泊松方程求解消除了压力的空间振荡。为了进一步精确地追踪物质界面的位置，本文通过离散质点生成全场的水平集函数，结合虚拟流体单元法可以精确施加表面张力的压力边界条件。针对半交错网格可能存在的虚假速度模态问题，本文采用一种类似有限元法中的沙漏阻尼对其进行了抑制。此外，通过预处理共轭梯度算法求解压力泊松方程提高了求解效率。针对不可压物质点法中存在的粒子凝聚现象，通过分析发现流体单元格心处满足散度为零条件，而在质点处是不满足的。为了解决这一问题，在流体单元满足散度为零的基础上，补充实现了流体质点的密度不变条件，提出了一种质点位置修正方案。新的改进不可压物质点法极大地降低了单元质点个数的要求，实现了流体质点的空间均匀化分布，保证了不可压流体体积的守恒性。为了解决物质点法和不规则运动刚体的耦合问题，基于能量最小原理的算子分裂算法提出了增广不可压物质点法。在此算法中，流体压力在不规则的运动刚体边界做的功被当做系统能量的一部分，通过变分原理可得到一个体积加权的压力泊松方程。利用新算法，本文研究了液体的小幅和大幅度晃动问题，并重点研究了挡板高度和个数对晃动的阻尼效应，通过研究最大晃动液面高度参数得到了挡板高度的最优值，为工程上的反晃动设计提供了参考。最后，为了实现不可压流体和变形体的流固耦合，本文提出了一种耦合不可压物质点有限元法。通过背景网格实现不可压流体质点和有限元结点之间的速度协调条件，通过多时间步算法保证了整体求解的时间效率。此外，本文通过穿透接触力来修正穿入有限元网格的流体质点，进而保证界面非穿透条件的实现。
Fluid flow with free surface has been widely applied in aerospace engineering, ship and ocean engineering and hydraulic engineering. It involves nonlinear hydrodynamics model as well as the change of nonlinear geometric topology of free surface such as splash, merge, fragmentation and overturning wave of liquid. The grid-based numerical methods have encountered huge challenges and difficulties in terms of the large derfomation of flow, tracking the free surface and fluid-structure interaction. However, the weakly compressible material point method (WCMPM) with the artificial equation of state has also encountered some issues including low efficiency and significant pressure oscillations. It's also hard to apply the pressure boundary condition at the free surface in WCMPM. To overcome these difficulties, a fully incompressible material point method (iMPM) is proposed in this paper for free surface flow. This new scheme uncouples the pressure term and the velocity term in the solution of N-S equation by the operator splitting technique, which also increases the size of time step in the explicit time integration significantly. The pressure of fluid is solved by the pressure Poisson equation that is able to eliminate the spatial pressure oscillations of the fluid.To track the free surface accurately, level set function is created in the whole field by discrete particles. By incorporating the ghost fluid method into the iMPM, the pressure boundary condition including the surface tension can be applied precisely. To suppress the spurious velocity modes existed in the semi-staggered grid, the hourglass damping which is analogous to the hourglass damping used in the finiteelement method is employed. Moreover, to ensure the efficiency of the method, pressure Poisson equation is solved by the preconditioned conjugate gradient (PCG) solver.The divergence-free condition is satisfied at the center of fluid cell but not satisfied at each fluid particle in the iMPM, so the unphysical particle clustering and voids appear in the iMPM. The density invariant condition is imposed at each fluid particle as a supplement of divergence-free condition, and a new particle shifting scheme is presented. This improved incompressible material point method reduces the number of particles per cell greatly, makes the distrubition of particles be order and uniform in the space and ensure fluid volume conservation.To handle the interaction between the incompressible fluid and movingsolid body with irregular boundary, an augmented incompressible material point method (AiMPM) is proposed based on the energy minimization form of the operator splitting technique. In this new scheme, the moving solid body with irregular boundary can be taken into account via the work done by the fluid pressure on the solid as a part of total energy by fluid pressure. According to the variational principle, the volume-weighted pressure Poisson equation is obtained. With the AiMPM, the liquid sloshing problems including the small-amplitude and large-amplitude were modeled, the investigation on the number and the height of middle baffles for the effects of sloshing damping is focused on. The optimal values of the height can be obtained through the curve of the most sloshing wave height (MSWH) which will supply a guidance for the design of anti-sloshing in engineering practice. Eventually, to realize the interaction between the incompressible fluid and the deformable solid, a coupled incompressible material point finite element method (CIMPFEM) is proposed, in which the velocity consistent condition between the incompressible fluid particles and the finite element nodes is achieved by the background grid and the multi-time step scheme is used to guarantee the computational efficiency. Besides, the penetration contact force is applied to correct the penetrating particles for the non-penetrating condition at the interface.
