In this work, a Lagrangian finite pointset model (FPM) is first developed to solve the 2D viscoelastic fluid governing equations, and then a corrected particle shifting technique (CPST) is introduced to eliminate the tensile instability in the long time simulations and added in the above FPM scheme (named as the FPMT). Subsequently, a coupled particle method (FPMT-SPH) is tentatively proposed to simulate the viscoelastic free surface flow, in which the FPMT method is used in the interior of fluid domain and the SPH method is adopted to treat the free surface near the boundary. The proposed FPMT method and FPMT-SPH method are motivated by: a) the spatial derivatives of the velocity and the viscoelastic stress are approximated and obtained by the weighted least squares method; b) the pressure is accurately solved by the application of projection method with a local iterative procedure; c) a corrected particle shifting technique is added to remedy the tensile instability (FPMT); d) an identification technique of free-surface particles is given in the FPMT-SPH method. The accuracy and the convergence of the proposed FPMT scheme for viscoelastic flow are first discussed by solving the planar flow based on the Oldroyd-B model, and compared with the analytical solutions. Secondly, the validity of the CPST is tested by several benchmarks, and compared with other numerical results. Thirdly, the viscoelastic lid-driven cavity flow at high Weissenberg number is simulated and compared with grid-based results, for further illustrating the robustness and the ability of the proposed FPMT method. Finally, the proposed coupled FPMT-SPH method is used to simulate the challenging free surface flow problem of a viscoelastic droplet impacting and spreading on rigid. All the numerical results show that the proposed particle method for the viscoelastic fluid or free surface flow is robust and reliable.

在这项工作中，首先开发了拉格朗日有限点集模型（FPM）来求解二维粘弹性流体控制方程，然后引入了修正的粒子移动技术（CPST）来消除长时间模拟中的拉伸不稳定性，并将其添加到高于FPM方案（称为FPMT）。随后，尝试提出了一种耦合颗粒法（FPMT-SPH）来模拟粘弹性自由表面流，其中在流体域内部使用FPMT方法，并采用SPH方法处理边界附近的自由表面。提出的FPMT方法和FPMT-SPH方法的动机是：a）速度和粘弹性应力的空间导数通过加权最小二乘法近似并获得；b）通过采用局部迭代程序的投影方法准确地解决了压力问题；c）增加了校正后的粒子移动技术以纠正拉伸不稳定性（FPMT）；d）FPMT-SPH方法给出了自由表面颗粒的识别技术。首先通过基于Oldroyd-B模型求解平面流，讨论了所提出的FPMT粘弹性流方案的精度和收敛性，并与解析解进行了比较。其次，CPST的有效性通过多个基准进行测试，并与其他数值结果进行比较。第三，模拟高Weissenberg数下的粘弹性盖驱动腔流动并将其与基于网格的结果进行比较，以进一步说明所提出的FPMT方法的鲁棒性和能力。最后，提出的FPMT-SPH耦合方法用于模拟粘弹性液滴冲击并散布在刚性表面上的具有挑战性的自由表面流动问题。所有数值结果表明，所提出的用于粘弹性流体或自由表面流的粒子方法是鲁棒且可靠的。