变节点余能原理基面力元法单元模型性能研究  

Performances of Variable-Node Elements With the Base Force Element Method Under the Complementary Energy Principle

作　　者：王耀[1] 胥民尧 纵岗[1] 侯长超 WANG Yao;XU Minyao;ZONG Gang;HOU Changchao(School of Architecture and Engineering,Yancheng Polytechnic College,Yancheng,Jiangsu 224005,P.R.China)

机构地区：[1]盐城工业职业技术学院建筑工程学院,江苏盐城224005

出　　处：《应用数学和力学》2025年第3期353-370,共18页Applied Mathematics and Mechanics

基　　金：2023年度国家外国专家项目(个人类)(G2023014042L);江苏省外国专家工作室项目;江苏省高职院校教师访学研修项目(2024GRFX071);江苏高校“青蓝工程”优秀青年骨干教师项目;江苏省高等学校自然科学研究A类项目(19KJB560006)。

摘　　要：为了解决疏密单元网格交界面节点位移不协调、求解方程构造复杂及空间可扩展性能差的问题,基于余能原理基面力元法提出了一种可变节点数量及位置的单元模型,并针对任意单元类型建立了一种具有统一形式的显式求解方法.首先,建立了一种二维可变边中节点单元模型,介绍了边中节点的柔度贡献矩阵及节点位移显式表达式;随后,将该模型扩展到三维层次,建立了一种可变面中节点单元模型,并将节点柔度贡献矩阵及节点位移表达式扩展到三维单元.基于上述模型,建立了平面及空间问题的疏密网格悬挂单元模型,并通过端部承受弯矩荷载、集中荷载及拉伸荷载的悬臂梁算例,论证了平面及空间可变节点单元模型、疏密网格悬挂单元模型的数值精度和适用性.研究表明,基于余能原理基面力元法建立的平面及空间可变节点单元模型具有较高的数值精度;此外,疏密网格之间的交界面无需进行任何处理措施,也无需构造插值函数或约束函数,仅依靠疏密网格交界面共用的边中节点(2D)及面中节点(3D)即可确保疏密网格交界面的节点位移协调;同时,模型和方法独立于单元类型、单元维度、节点数量及分布等因素,具有优异的空间可扩展性能及易于程序化的特点.To address the problems of uncoordinated nodal displacements at the intersection of the sparse and dense elements, complicated construction of solving equations, and poor spatial scalability, an element model with variable node numbers and positions was proposed with the base force element method(BFEM) under the complementary energy principle, and an explicit solution method in a unified form was established for arbitrary element types. First, a 2D variable mid-edge node element model was established, and the explicit expressions of the contribution of nodes compliance matrix and nodal displacements were introduced. Subsequently, the element model was extended to the 3D form, a variable mid-face node element model was proposed, and the above expressions were also extended to the 3D forms. Hereafter, a sparse and dense mesh hanging element model was established, and the numerical accuracy and applicability of the planar and spatial variable-node element model were demonstrated with the cantilever beam subjected to a bending moment load, a concentrated load, and a tensile load at the end. The numerical results show that, the variable-node element model and the hanging element model in the 2D and 3D forms based on the BFEM have high numerical accuracy. In addition, the nodal displacement coordination at the intersection interface between sparse and dense elements can be ensured only by the shared mid-edge node(2D) and the mid-face node(3D) at the interface without any processing treatment or construction of interpolation functions and constraint functions. Meanwhile, the element models and methods are independent of the element type, element dimensions, nodes' number, and nodes' distribution, etc., and have excellent spatial scalability and programmability.

关 键 词：基面力元法 变节点单元 余能原理 悬挂单元模型 

分 类 号：O343.1[理学—固体力学] O343.2[理学—力学]