本项目针对不可压缩黏弹性两相流模拟中的两个主要问题：黏弹性输运流体流动模拟算法，以及液相、固相交界面的捕捉方法，开展相应研究并提供有效解决方案。首先，基于有限增量理论（FIC）在黏弹性流动控制方程中引入稳定化机制，保证其对初边值条件的适定性，绕开相容性条件对混合变量插值空间的限制，并抑制微分型黏弹性本构方程中的对流占优问题。在此基础上构建出"低阶高精度"的求解方案。其次，利用扩展有限元（XFEM）建立不依赖于有限元单元拓扑构型的移动界面捕捉方法。移动界面不需要与单元边界重合, 避免了频繁的网格广顺和网格重构，并且能准确预测黏弹性两相流的场变量（速度、压力、黏弹应力等）在颗粒表面的间断性。在此基础上充分利用无网格法高阶逼近特性以及其自适应优势，构建"有限元-无网格"耦合的黏弹性两相流模拟求解方案，为颗粒附近流场、具有高阶梯度性的应力场提供高精度求解策略。

Two main issues in numerical simulation of incompressible viscoelastic two-phase flows are investigated in this project, i.e., study on algorithms for the simulation of viscoelastic fluid acting as transport media in two-phase flows, and study on interface capturing methods for the interface formed by viscoelastic transport fluid and particles. Firstly, Based on theory of finite calculus increment, stabilization mechanisms are incorporated into governing equations to ensure the well-posedness to the initial and boundary conditions, to circumvent the compatibility condition imposed by the incompressibility for interpolations of mixed variables, and to depress convection domination in the differential viscoelastic constitutive equation. Furthermore, the solution algorithm is constructed, with the property of obtaining results with high accuracy by lower-order interpolations. Secondly, by virtue of extended finite element method, the interface capturing method is constructed without the necessity of considering the connections of finite elements. By this method, the alignment of the elements' boundaries and moving interfaces are not necessary, leading to computational costs reduction due to no mesh smoothing and remeshing required, meanwhile the discontinuities across surfaces of particles are predicted accurately for the field variables such as velocity, pressure and viscoelastic stress, etc. On the basis as accounted above, the coupled meshfree-finite element method is developed to provide high resolutions to velocity field and stress field with high stress gradient near the moving particles, by virtue of the advantage on higher order interpolations and self-adaptivity of meshfree method.