A stable SPH discretization of the elliptic operator with heterogeneous coefficients Abstract   Smoothed particle hydrodynamics (SPH) has been extensively used to model high and low Reynolds number flows, free surface flows and collapse of dams, study porescale flow and dispersion, elasticity, and thermal problems. In different applications, it is required to have a stable and accurate discretization of the elliptic operator with homogeneous and heterogeneous coefficients. In this paper, the stability and approximation analysis of different SPH discretization schemes (traditional and new) of the diagonal elliptic operator for homogeneous and heterogeneous media are presented. The optimum and new discretization scheme of specific shape satisfying the twopoint flux approximation nature is also proposed. This scheme enhances the Laplace approximation (Brookshawxe2x80x99s scheme (1985) and Schwaigerxe2x80x99s scheme (2008)) used in the SPH community for thermal, viscous, and pressure projection problems with an isotropic elliptic operator. The numerical results are illustrated by numerical examples, where the comparison between different versions of the meshless discretization methods are presented.

A leastsquares particle model with other techniques for 2D viscoelastic fluidfree surface flow Abstract   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 (FPMTSPH) 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 FPMTSPH 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 freesurface particles is given in the FPMTSPH 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 OldroydB 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 liddriven cavity flow at high Weissenberg number is simulated and compared with gridbased results, for further illustrating the robustness and the ability of the proposed FPMT method. Finally, the proposed coupled FPMTSPH 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.