中文摘要：

在有限增量微积分（finite increment calculus，FIC）的理论框架下，通过引入一个附加变量，发展了压力稳定型分步算法，有效改善了经典分步算法的压力稳定性，同时还避免了标准FIC方法中存在的空间高阶导数的计算．为保证数值方法同时具有较快的计算速度和较好的健壮性，发展了有限元与无网格的耦合空间离散方法．该方案可在网格发生扭曲的区域采用无网格法空间离散以保证求解的精度和稳定性，而在网格质量较好的区域以及本质边界上保留使用有限元法空间离散以提高计算效率和便于施加本质边界条件．方腔流考题的数值模拟结果突出地显示了所发展的压力稳定型分步算法比经典分步算法具有更好的压力稳定性，能够有效消除速度一压力插值空间违反LBB条件而导致的压力场的虚假数值振荡．平面Poisseuille流动和一个典型型腔充填过程的数值模拟结果，表明了发展的耦合离散方案相对于单一的有限元法和单一的无网格法在综合考虑计算效率和算法健壮性方面的突出优点．

英文摘要：

By introducing an additional variable in the framework of the Finite Increment Calculus（FIC） theory, a pressure stabilized fractional step algorithm is developed in this paper with enhanced pressure stability in comparison with the classic one. In the algorithm, the calculation of the high order spatial derivatives as required in the standard FIC procedure is avoided. To ensure superior overall performance of the proposed numerical scheme in accuracy, efficiency and robustness, a coupled finite element and meshfree method is developed for the spatial discretization and interpolation approximation, in which the meshfree approximation is adopted in the region where the mesh is distorted to preserve the accuracy and robustness of numerical solutions, while the finite element approximation is employed in the region where the quality of the mesh is acceptable and on the boundaries where essential boundary conditions of flow problems are imposed to ensure high computational efficiency and proper imposition of the essential boundary conditions. Numerical results for the lid-driven cavity flow problem demonstrate that the pressure stability of the proposed pressure stabilized fractional step algorithm is better than that of the classic one, and the algorithm can get rid of spurious oscillations in the resulting pressure field induced by the incompatible interpolation approximations for the velocity and pressure fields in violating the LBB condition. Two example problems, i.e. the plane Poisseuille flow and the injection molding problems are calculated to demonstrate the superiority of the proposed coupled finite element and meshfree method over either the finite element or meshfree methods in the overall performance.