Lagrangian meshfree finite difference particle method with variable smoothing length for solving wave equations

Sheng Wang,Jing Ping Wu,Yong Ou Zhang

doi:10.1177/1687814018789248

Sheng Wang, Jing Ping Wu + Show 1 more

Open Access

https://doi.org/10.1177/1687814018789248

Copy DOI

Abstract

In a Lagrangian meshfree particle-based method, the smoothing length determines the size of the support domain for each particle. Since the particle distribution is irregular and uneven in most cases, a fixed smoothing length sometime brings too much or insufficient neighbor particles for the weight function which reduces the numerical accuracy. In this work, a Lagrangian meshfree finite difference particle method with variable smoothing length is proposed for solving different wave equations. This pure Lagrangian method combines the generalized finite difference scheme for spatial resolution and the time integration scheme for time resolution. The new method is tested via the simple wave equation and the Burgers’ equation in Lagrangian form. These wave equations are widely used in analyzing acoustic and hydrodynamic waves. In addition, comparison with a modified smoothed particle hydrodynamics method named the corrective smoothed particle method is also presented. Numerical experiments show that two kinds of Lagrangian wave equations can be solved well. The variable smoothing length updates the support domain size appropriately and allows the finite difference particle method results to be more accurate than the constant smoothing length. To obtain the same level of accuracy, the corrective smoothed particle method needs more particles in the computation which requires more computational time than the finite difference particle method.

Highlights

Numerical methods have been widely implemented to model acoustic and hydrodynamic waves
Considering the generalized finite difference (GFD) scheme for spatial derivatives with arbitrarily distributed points, we proposed a Lagrangian meshfree finite difference particle method (FDPM) for solving wave equations
W where W(hj,rs) and W(hj,kj,rs) are the weight functions in one and two dimensions, respectively, and rs represents the size of the support domain, which is always called the smoothing length in the smoothed particle hydrodynamics (SPH) method

Summary

Introduction

Numerical methods have been widely implemented to model acoustic and hydrodynamic waves. Lagrangian meshfree particle-based approach for solving different time-dependent wave equations. Considering the generalized finite difference (GFD) scheme for spatial derivatives with arbitrarily distributed points, we proposed a Lagrangian meshfree finite difference particle method (FDPM) for solving wave equations. Zhang and Batra[31,32] have proposed the modified smoothed particle method (MSPH) and the symmetrical smoothed particle hydrodynamics (SSPH) method with a higher order accuracy These methods used the Taylor series expansion to modify the kernel function, the FDPM still have its advantages. Qiang and Gao[38] implicitly coupled the density evolution equation with variable smoothing length to deal with the large density gradient and large smoothing length gradient problems All these ways to change the smoothing length are proposed with their own pros and cons, and they need to use the physical density to update the length, which is not suitable for this work. The sum of these expressions for all particles j with multiplying the weight functions is obtained

CCCCCCCCA

Conclusion