A fully Lagrangian mixed discrete least squares meshfree method for simulating the free surface flow problems

This paper presents a fully Lagrangian mixed discrete least squares meshfree (MDLSM) method for simulating the free surface problems. In the proposed method, the mass and momentum conservation equations are first discretized in time using the projection method. The resulting pressure Poisson equation is then re-written in the form of three first-order equations in terms of the pressure field and its first-order derivatives. The mixed discrete least squares meshless method is then used to solve this system of equations and simultaneously calculate the pressure field and its gradients. The advantage of the proposed Lagrangian MDLSM is twofold. First, the pressure gradients are directly computed and, therefore, they enjoy higher accuracy than those calculated in the conventional DLSM via a post-processing method. The more accurate pressure gradient will in turn lead to more accurate velocity field when used in the momentum equations. Second, the proposed Lagrangian MDLSM method can be more efficient than the corresponding original Lagrangian DLSM method, for the specific number of nodes, since the costly calculation of the shape function second derivatives required for solving the pressure Poison equation are avoided in each time step of the simulation. Several free surface problems are solved and the results are presented and compared to those of DLSM method. The results indicate the superior efficiency and accuracy of the proposed Lagrangian MDLSM method compared to those of the existing Lagrangian DLSM method in the literature.