# Point Cloud Movement For Fully Lagrangian Meshfree Methods

## TL;DR

This paper introduces advanced methods for moving point clouds in meshfree Lagrangian simulations, significantly improving accuracy and conservation properties over traditional first order approaches.

## Contribution

It proposes novel movement algorithms based on characteristic velocities, extending streamline tracing and pathline approximation techniques to meshfree methods.

## Key findings

- New methods outperform first order movement in accuracy
- Significantly reduce volume and mass conservation errors
- Numerical tests demonstrate superior performance of proposed methods

## Abstract

In Lagrangian meshfree methods, the underlying spatial discretization, referred to as a point cloud or a particle cloud, moves with the flow velocity. In this paper, we consider different numerical methods of performing this movement of points or particles. The movement is most commonly done by a first order method, which assumes the velocity to be constant within a time step. We show that this method is very inaccurate and that it introduces volume and mass conservation errors. We further propose new methods for the same which prescribe an additional ODE system that describes the characteristic velocity. Movement is then performed along this characteristic velocity. The first new way of moving points is an extension of mesh-based streamline tracing ideas to meshfree methods. In the second way, the movement is done based on the difference in approximated streamlines between two time levels, which approximates the pathlines in unsteady flow. Numerical comparisons show these method to be vastly superior to the conventionally used first order method.