独立覆盖流形法的通用计算公式和通用程序设计——(一)通用计算公式  

General Formulas and Program Design for Manifold MethodBased on Independent CoversⅠ:General Formulas

作　　者：苏海东[1,2] SU Hai-dong(Material and Engineering Structure Department,Changjiang River Scientific Research Institute,Wuhan 430010,China;Research Center of Water Engineering Safety and Disaster Prevention of Ministry ofWater Resources,Wuhan 430010,China)

机构地区：[1]长江科学院材料与结构研究所,武汉430010 [2]水利部水工程安全与病害防治中心,武汉430010

出　　处：《长江科学院院报》2025年第4期193-201,210,共10页Journal of Changjiang River Scientific Research Institute

基　　金：国家自然科学基金项目(U2340229)。

摘　　要：独立覆盖流形法是偏微分方程数值计算的新方法,在近似函数构造的基础层面上形成了偏微分方程的“分区级数解”,实现了有限元法及一些新方法的主要分析功能,在网格划分的灵活性、计算稳定性等方面还具有一些独特优势,但也意味着其计算公式和程序设计不同于现有方法。总结了近年来该方法在固体计算中的主要研究成果,归纳了一套简洁的通用计算公式,局部近似函数中的形函数表达为单位分解函数、坐标转换矩阵和级数矩阵的乘积,具体讨论了各种情况下的形函数及其求导方法,给出各种矩阵的表达式以及时间积分方法,用于求解弹性力学运动微分方程、传导方程、波动方程,包括一维至三维的稳态和瞬态分析和三类边界条件,涵盖了高阶级数、任意形状网格、精确几何边界模拟及本质边界条件的准确施加、裂纹尖端附近采用解析级数等特色功能。利用这些公式即可开展独立覆盖流形法通用程序的开发工作。Manifold method based on independent covers is a novel approach for numerically solving partial differential equations.By constructing approximate functions,it generates a“partitioned series solution”for partial differential equations.This method not only achieves the main functions of the finite element method(FEM)and other numerical techniques but also outperforms them in certain aspects,such as mesh generation flexibility and computational stability.However this also means that its calculation formulas and program design are different from existing methods.This paper reviews the major research outcomes in solid computation in recent years,and summarizes a set of simple and general calculation formulas in which the shape function of the local approximation function is expressed as the product of the Partition of Unity(PU)function,coordinate transformation matrix,and series matrix.The shape function and its derivatives under various scenarios are discussed in details.Different matrices and the time integration method are also given.These formulas can be applied to solve the differential equations of motion in elasticity,conduction equations,and wave equations,covering one-to-three-dimensional steady-state and transient analyses,along with three types of boundary conditions.They offer features such as high-order series,arbitrary mesh shapes,accurate boundary geometric simulation,precise application of essential boundary conditions,and local analytical series near the crack tip.Utilizing these formulas,a general program for the new method can be developed.

关 键 词：偏微分方程 级数解 网格剖分 精确几何 独立覆盖 数值流形方法 

分 类 号：TB115[理学—数学]