二维不可压材料线弹性有限体积法的研究

Two-dimensional incompressible linear elasticity analysis by finite volume method

摘要:

发展了一种针对二维不可压材料线弹性问题的格点型有限体积法(CV-FVM)，将不可压约束条件加入到线弹性体控制方程中，采用双线性四边形单元，对控制方程的数值离散过程做了详细的推导. 将位移以及静水压力作为待解量进行直接求解，其分别存储在单元节点以及单元中心，并假设静水压力在单元内分布一致. 对不可压材料的方板以及无限长圆管的线弹性问题进行了数值验证，通过数值解与理论解的对比，发现采用发展的CV-FVM可以避免有限元中当泊松比为0.5时所出现的闭锁现象，通过增加网格密度可以得到较高精度的数值解.

Abstract:

A novel two-dimensional finite volume procedure is described in detail for the analysis of incompressible linear elastic body, the volume conservation for each volume form a constrain equation is added to governing equation of linear elastic body, the procedure of discretize governing equations using bi-linear quadrilateral elements is given in detail. The discrete equation contains displacement and hydrostatic pressure as unknown variable, and the hydrostatic pressure is storied at the center of the element and assumed to be constant within each element. The developed procedure is tested on two linear elastic plane strain benchmark problems of impressible material, and then compared with analytical results. In addition, no locking problem is encountered as the poisson’s ratio equal to 0.5 which is a common problem in FEM, the simulation results show that this FVM can obtain an accurate and stable numerical solution under mesh refinement.