A numerical method for the solid mechanics with Euler variables

Junchen Liu,Feng Qian,Junfu Wang,Chun Feng,Shihai Li,Ling Wang,Xinguang Zhu

doi:10.1088/1742-6596/1980/1/012018

Abstract

Based on the requirement of field engineering, we proposed a new numerical method for analyzing the solid mechanics with Euler variables. By replacing the Lagrange variables with the Euler variables, there will be three new items appearing in the equation. We use the fixed Euler grid to calculate solid mechanics, and use the boundary of material regiment to describe the movement of the boundary of material, after the boundary of material regiment moves beyond the Euler element grid size, new Euler element grids will appear and the corresponding old Euler element grids which have been out of the region of material will disappear. By this way, the movement of solid could be calculated in the Euler fixed grids. Then we did some test cases, we get the result that the new method is right and could be used to calculate the solid mechanic problems.

Highlights

In the research, we compute the fluid mechanics with Euler variables, and the solid mechanics with Lagrange variables
By replacing the Lagrange variables with the Euler variables, there will be three new items appearing in the equation
We use the fixed Euler grid to calculate solid mechanics, and use the boundary of material regiment to describe the movement of the boundary of material, after the boundary of material regiment moves beyond the Euler element grid size, new Euler element grids will appear and the corresponding old Euler element grids which have been out of the region of material will disappear

Summary

Introduction

We compute the fluid mechanics with Euler variables, and the solid mechanics with Lagrange variables. Liu Chuanqi and his colleagues [11] have done a thorough research on this problem ,they proposed an implicit material point method designed to bypass meshing of bodies by employing level set functions to represent the boundaries at Euler grids. This implicit function representation provides an elegant mean to link an unbiased intermediate reference surface with the true boundaries by closest point projection as shown in Leichner et al (2019). The movement and the deformation of the solid material could be calculated in the Euler fixed grids

Discussion

The first item By introducing

Conclusions