方程组降阶算法及其在电工装备数值计算中的应用  

Equation System Order-Reduced Algorithms and Their Application in Numerical Computation for Electrical Equipment

作　　者：金军 阎秀恪[1] 钟立国 张艳丽[1] 任自艳[1] Jin Jun;Yan Xiuke;Zhong Liguo;Zhang Yanli;Ren Ziyan(School of Electrical Engineering,Shenyang University of Technology,Shenyang,110870,China)

机构地区：[1]沈阳工业大学电气工程学院,沈阳110870

出　　处：《电工技术学报》2025年第7期2020-2032,共13页Transactions of China Electrotechnical Society

基　　金：国家自然科学基金项目(52277015);辽宁省研究生教育教学改革研究项目(LNYJG2022060)资助。

摘　　要：大型电气设备在进行有限元分析时,会遇到大规模数值计算问题。代数方程组阶数巨大导致计算时间长,计算效率低,有时甚至无法求解。该文研究有限元代数方程组的降阶求解,推导了分块迭代算法、正交降阶分解算法(OORDA),将禁忌搜索算法引入高斯消元法形成改进的高斯消元法(IGEM),提出了分块迭代算法分别与OORDA和IGEM相结合的方程组混合降阶算法,并通过编程实现。将OORDA、IGEM和两种混合降阶算法分别应用于长直载流导体的磁场有限元计算中,计算结果与解析解的对比验证了算法的正确性。将四种算法应用到单相变压器的磁场有限元计算中,两种混合降阶算法能够快速大幅降低方程组阶数,提高计算效率。该文提出的降阶算法可应用于电工装备物理场的大规模数值计算中,能显著提高复杂模型的有限元计算效率。The development of digital twin technology for electrical equipment requires extensive calculation.For the electrical equipment with large physical dimension or complicated structure,a large number of meshing grids are needed to analyze its physical field accurately using finite element method(FEM),which leads to a huge order of algebraic equations and makes the solution very difficult,and sometimes it cannot be solved.For sparse symmetric equations obtained by finite element analysis,this paper researches effective order-reduced algorithms.Firstly,the block iterative algorithm is derived and explored in detail,in which the block diagonalization method is proposed to transform the finite element stiffness matrix into the diagonal matrix.The diagonal matrix is the matrix with unit matrix at diagonal,and the matrix is divided into sub-matrix blocks according to the unit matrix.Next,the block left multiplication method is proposed to realize the movement of sub-matrix blocks and block reorganization,and the permutation matrix method is also derived to realize the process.The block reorganization can reduce the order of the equations to half of its original order,and can reduce the order repeatedly.Taking the finite element analysis of the magnetic field of the open slot of the motor as an example,the block iterative algorithm is used twice to reduce the order of the equations,and the original 10th-order stiffness matrix is reduced to 3rd order.Next,the mathematical model of orthogonal order-reduced decomposition algorithm(OORDA)is also derived.OORDA is to orthogonalize any column of the matrix with other column vectors,removes elements from corresponding row and column,and the matrix will be reduced by one order.This operation is repeated until the matrix is reduced to first-order,and then substitute the result back to complete the solution.Aiming at the characteristics of sparsity of finite element stiffness matrix,the taboo search algorithm is introduced into the Gaussian elimination method to form an improved Gau

关 键 词：有限元分析 代数方程组 降阶算法 大规模数值计算 计算效率 

分 类 号：TM154[电气工程—电工理论与新技术]