基于辛体系求解板壳结构的面相似比例边界有限元法研究  

STUDY ON SCALED BOUNDARY FINITE ELEMENT METHOD FOR PLATES AND SHELLS UPON SCALING FACE AND HAMILTON SYSTEM

作　　者：李云途 刘小楠 胡志强 阚宇彤 李炳奇[1] LI Yun-tu;LIU Xiao-nan;HU Zhi-qiang;KAN Yu-tong;LI Bing-qi(State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin,China Institute of Water Resources and Hydropower Research,Beijing 100038,China;Faculty of Infrastructure Engineering,Dalian University of Technology,Dalian,Liaoning 116024,China;China Academy of Building Research,Beijing Glory PKPM Technology Co.,Ltd.,Beijing 100013,China)

机构地区：[1]流域水循环模拟与调控国家重点实验室,中国水利水电科学研究院,北京100038 [2]大连理工大学建设工程学部,辽宁大连116024 [3]中国建筑科学研究院有限公司北京构力科技有限公司,北京100013

出　　处：《工程力学》2025年第4期46-55,共10页Engineering Mechanics

摘　　要：针对板壳结构的数值模拟,基于辛体系建立了面相似比例边界有限元法(SBFEM)。区别于经典比例边界有限元法以点为相似中心,面相似比例边界有限元法以板壳结构环向边界面为比例变换中心,对板壳的表面应用二维谱单元进行离散,建立表面单元与比例中心面的变换关系。基于哈密顿体系理论推导了以面为相似中心的控制方程。与此同时,针对以局部径向参数为变量的系数矩阵,进一步采用二次多项式拟合确定方程组的哈密顿系数矩阵,得到了板壳结构的表面刚度阵和节点位移函数表达形式。面相似SBFEM基于三维实体理论进行建立和求解,有效地解决了传统实体有限单元在模拟板壳时存在的剪切闭锁问题。提出的方法适用于任意形状和变厚度的板壳结构。通过经典数值算例,验证了所建立模型的计算精度和适用性。Based on a Hamiltonian system,a scaled boundary finite element method(SBFEM)is established for the numerical simulation of plates and shells.Unlike the classical SBFEM,which takes a point as the scaling center,the method proposed takes a circumferential boundary face as the transformation center of the plates and shells.And,in the novel SBFEM,the surface of structures is discretized with two-dimensional spectral elements.The transformation relationship between the boundary face and the scaling face are established,and the governing equations with the scaling face based on the Hamiltonian system are derived.At the same time,for the coefficient matrix with local radial parameter as the variable,the Hamiltonian coefficient matrix of the equation is determined by quadratic polynomial fitting,and the expressions of the stiffness matrix and the nodal displacement function of the structures are obtained.The proposed SBFEM based on scaling face is established and solved upon three-dimensional solid theory,which effectively overcomes the locking problem of traditional finite element in the plates and shells.The method proposed is suitable for plates and shells with varying thickness and arbitrary shapes.The accuracy and correctness of the model are verified by classical numerical examples.

关 键 词：比例边界有限元法 面相似 板壳结构 哈密顿体系 谱单元 

分 类 号：O343[理学—固体力学]