有限元法中间构形初始解预示的Laplace-Beltrami方程法  

A METHOD OF LAPLACE-BELTRAMI EQUATIONS FOR THE INITIAL GUESS SOLUTION OF INTERMEDIATE CONFIGURATIONS IN FEM

作　　者：刘永财 陈文亮 胡庆婉 鲍益东 LIU Yong-cai;CHEN Wen-liang;HU Qing-wan;BAO Yi-dong(College of Mechanical and Electrical engineering,Nanjing University of Aeronautics&Astronautics,Jiangsu Province Key Laboratory of Precision and Micro Manufacturing Technology,Nanjing 210016,China;Department of Mathematics and Statistics,Qujing Normal College,Qujing 655011,China)

机构地区：[1]南京航空航天大学机电学院,江苏省精密与微细制造技术重点实验室,南京210016 [2]曲靖师范学院数学与统计学院,曲靖655011

出　　处：《工程力学》2020年第6期42-50,共9页Engineering Mechanics

基　　金：国家自然科学基金项目(51875283);航空科学基金项目(20161852018);江苏省精密与微细制造技术重点实验室2018年开放基金项目(2018054)。

摘　　要：一步有限元法具有高效的计算能力,在板料成形模拟方面得到了广泛的应用,但它存在应力计算精度不足的问题。为弥补这一不足,通过增加中间构形的方式,人们提出多步有限元法。而中间构形需根据初始解迭代计算获得,因此,初始解预示算法是多步有限元法的一个关键问题。该文把中间构形解耦分解为弯曲变形和拉伸变形两个独立的变形过程,且将弯曲变形作为中间构形的初始解预示,改善了多步有限元法的稳定性;并根据大位移小应变理论,建立了节点的位移约束条件;进一步,首次通过Laplace-Beltrami方程的建立和求解,实现了中间构形初始解的快速预示,该方法易于编程实现,稳定性好。通过典型零件数值算例的高效准确计算,验证了该算法的可行性和有效性。The one-step finite element method(FEM)has been applied widely in metal forming simulation because it can obtain solutions quickly.However,the problem of poor accuracy for stress may rise in this approach.By means of adding several intermediate configurations,a multi-step FEM is proposed to overcome this disadvantage.The intermediate configurations can be acquired by an iterative algorithm from an initial guess,and a desired initial guess of the intermediate configuration is very important for the multi-step FEM.In light of the decoupled method,an intermediate configuration is composed of two independent deformations,that is,the bending deformation and stretching deformation.The bending deformation is considered as the initial guess of the intermediate configuration so that the stability and convergence of the solution in the multi-step FEM are improved.According to the large displacement with small strain theory,the displacement constraints of nodes are established.The initial guess of the intermediate configurations is obtained accurately by the model of Laplace-Betrami equations for the first time.This method is easy to implement and has good stability.Numerical simulations of some standard stamping parts validate the effectiveness of the algorithm.

关 键 词：板料成形 多步有限元法 解耦格式 中间构形初始解 Laplace-Beltrami方程 

分 类 号：O241.82[理学—计算数学]