Simulating free surface problems using Discrete Least Squares Meshless method

Abstract

A Discrete Least Squares Meshless (DLSM) method is presented here for the simulation of incompressible free surface flows. The governing equations of the mass and momentum conservations are solved in a Lagrangian form using a pressure projection method. Since there are no particles in the outer region of the free surface, the particle density will drop significantly. Free surfaces are, therefore, resolved by tracking particles with highly reduced density. A fully least squares approach is used in both function approximation and the discretization of the governing differential equations in space. The meshless shape functions are derived using the Moving Least Squares (MLS) method of function approximation. The discretized equations are obtained via a discrete least squares method in which the sum of the squared residuals are minimized with respect to unknown nodal parameters. The method enjoys the advantage of producing symmetric, positive definite matrixes for the cases considered. The method can be viewed as a truly meshless method since it does not need any mesh for both field variable approximation and the construction of system matrices. Two free surface problems namely dam break and evolution of a drop with an initial known velocity are solved to test the accuracy of the proposed method. The results show the ability of the proposed method to solve complex fluid dynamic problems with moving free surface boundaries.