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| Equation System Order-Reduced Algorithms and Their Application in Numerical Computation for Electrical Equipment |
| Jin Jun, Yan Xiuke, Zhong Liguo, Zhang Yanli, Ren Ziyan |
| School of Electrical Engineering Shenyang University of Technology Shenyang 110870 China |
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Abstract 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 Gaussian elimination method(IGEM).The information of non-zero elements is extracted and recorded in the taboo information table, so as to eliminate the memory occupation and computation increase caused by zero elements during elimination and back-substitution processes, which will greatly reduce the number of operations and computational cost of Gaussian Elimination method.
Due to the notable memory requirements of both OORDA and Gaussian elimination method, and the fact that the block iterative algorithm can substantially reduce the order of the equations, the Hybrid Algorithm 1 (HA1) that combine the block iterative algorithm with OORDA and Hybrid Algorithm 2 (HA2) that combine the block iterative algorithm with IGEM are proposed in this paper. The four algorithms, OORDA, IGEM, HA1 and HA2 are applied to the finite element analysis of the magnetic field produced by a long straight wire carrying current, the correctness of the algorithms have been verified by comparing the calculation results with the analytical solution. The four algorithms are applied to calculate the magnetic field distribution of a single-phase transformer under different meshes. The results show that using the block iterative algorithm to reduce the order of the equations, and then employing the OORDA or IGEM to solve them can greatly improve the computational efficiency. The larger the calculation scale, the more significant the effect. The HA2 shows the highest computational efficiency. The OORDA is not suitable for solving high-order equations alone, but it is a feasible choice for matrices that are inconvenient to inverse.
All the algorithms in this paper are realized by programming in C++, and the order reduction algorithms can provide a solution approach and reference for large-scale numerical calculation.
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Received: 29 March 2024
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2025, Vol. 40 
