A new meshless approach in simulating free surface flows using continuous MLS shape functions and Voronoi diagram

The purpose of this paper is to develop an efficient and accurate mesh-less method to simulate free flows with continuous deformation in boundary positions.

A two-step pressure projection method in a Lagrangian form is used to solve the governing equations of mass and momentum conservation. In the first step, velocity field is calculated in which incompressibility is not enforced. In the second step, a pressure Poisson equation is applied to satisfy incompressibility conditions. The numerical proposed method is used for spatial discretization of the governing equations. Three benchmark-free surface problems, namely, dam break, solitary wave propagation and evolution of an elliptical bubble with available experimental results and analytical solutions, are used to test the accuracy of the proposed method. The results prove the accuracy of the method in simulating free surface problems.

The Voronoi diagram instead of kernel function summation can be used to estimate the particle or nodal volume concept in particle-based (mesh-less) methods for function approximation. This idea probably works well especially for highly irregular node distributions.

The continuous moving least squares shape functions are applied for function approximation, and the Voronoi diagram concept is also used to estimate region influence of computational nodal points or particle volumes. Combinations of these two concepts and finite differences formulation for first derivatives gives an accurate numerical model for Laplacian operator in the proposed method.