动力松弛法在应变软化类结构有限元静力分析中的应用

Application of dynamic relaxation method in finite element static-solution of strain-softening-type structure

结构静力分析中常因材料应变软化使得相应定解问题失去适定性,从而导致有限元分析不收敛。为解决此问题,在已有的相关研究基础上,采用动力松弛法(DRM)求解结构非线性有限元静力分析的增量步,将其应用于损伤型本构所描述的结构软化问题。本文方法依据两个原理,其一是苏联《数学百科全书》论述的原理——定义于时间域的任何定解问题适定可解,其二是DRM所用的原理——质量系统静力解为相应动力解的稳态部分。且DRM无需进行隐式静力分析时的总体刚度矩阵组装和求逆计算。本文用加荷载增量求解静力平衡路径硬化段,用加位移增量求解极值点和软化段。数值试验表明,本文方法能完成应变软化类结构的静力平衡路径求解。

In non-linear static analysis of structures with strain softening materials,the solution process often loses well-posedness of the problem,leading to divergence of the finite element analysis process.In order to solve this problem,we use the dynamic relaxation method(DRM)to solve the static increments in the incremental solution procedure of damage type softening structure by further developing relevant research,so that the solution process becomes convergent.Our method is based on two principles:the first principle is that any problem of definite solution defined on the time domain will be well-posed and solvable as described in the Encyclopedia of Mathematics published in former Soviet Union,and the second principle is that the static solution of a mass system can be the stable part of its dynamic solution as is the principle used in DRM.Moreover,in DRM there is no need to assemble and invert the stiffness matrix as in implicit-static-analysis such that the associated computational cost is removed.The ascending-branch of static equilibrium path is solved by load increments,while the peak-point and the descending-branch are solved by displacement increments.Two numerical examples demonstrate the effectiveness of such an application of DRM in the finite element analysis of static equilibrium path of strain-softening structures.