摘要
应变-旋转(Strain-Rotation,S-R)和分解定理为分析几何非线性问题提供了合理可靠的理论基础,但用有限元求解时会遇到大变形发生后的网格畸变问题。近年提出的虚单元法(Virtual element method,VEM)适用于一般的多边形网格,因此,该文尝试使用一阶虚单元求解基于S-R和分解定理的二维几何非线性问题,以克服网格畸变的影响。基于重新定义的多项式位移空间基函数,推演获得一阶虚单元分析线弹性力学问题时允许位移空间向多项式位移空间的投影表达式;按照虚单元法双线性格式的计算规则,分析处理基于更新拖带坐标法和势能率原理的增量变分方程;进而建立离散系统方程及其矩阵表达形式,并编制MATLAB求解程序;采用常规多边形网格和畸变网格,应用该文算法分析均布荷载下的悬臂梁和均匀内压下的厚壁圆筒变形。结果与已有文献和ANSYS软件的对比表明:该文算法在两种网格中均可有效执行且具备足够数值精度。总体该文算法为基于S-R和分解定理的二维几何非线性问题求解提供了一种鲁棒方法。
The strain-Rotation(S-R)decomposition theorem provides a reliable and reasonable theoretical basis for geometric nonlinear analyses.However,the mesh distortion is inevitable in a large deformation analysis when using the finite element method.The virtual element method(VEM)proposed recently is applicable to general polygonal meshes.In the study,the first order VEM is used to solve 2D geometric nonlinear problem upon S-R decomposition theorem,aiming to overcome the impact of mesh distortion.One project operator is derived for the linear elastic problem from the admissible displacement space to the polynomial displacement space,by redefining the basis function of the polynomial displacement space.The incremental variation equation based on the updated co-moving coordinate formulation and the potential energy rate principle is computed according to the computation rule of the bilinear form in VEM.The discretization equation is obtained with the matrix expression,and a solution program is coded in MATLAB.The proposed method is employed in the deformation analysis on a cantilever beam under distributed loads and on a thick cylinder under uniform internal pressures,by using a general polygonal mesh and a distorted mesh.Comparison on the results by the method proposed,and those from the published literatures and of ANSYS shows that the method proposed is valid regardless of the mesh distortion and,has a sufficient numerical precision.In conclusion,the method proposed is robust to solve 2D geometric nonlinear problems based on S-R decomposition theorem.
作者
江巍
尹豪
吴剑
汤艳春
李坤鹏
郑宏
JIANG Wei;YIN Hao;WU Jian;TANG Yan-chun;LI Kun-peng;ZHENG Hong(Key Laboratory of Geological Hazards on Three Gorges Reservoir Area,Ministry of Education,China Three Gorges University,Yichang,Hubei 443002,China;College of Civil Engineering&Architecture,China Three Gorges University,Yichang,Hubei 443002,China;College of Architecture and Civil Engineering,Beijing University of Technology,Beijing 100124,China)
出处
《工程力学》
EI
CSCD
北大核心
2024年第8期23-35,共13页
Engineering Mechanics
基金
国家自然科学基金项目(52079070)
三峡库区地质灾害教育部重点实验室开放基金项目(2020KDZ10)。
关键词
S-R和分解定理
虚单元法
几何非线性
网格畸变
多边形网格
S-R decomposition theorem
virtual element method
geometric nonlinearity
mesh distortion
polygonal mesh
