Electromagnetic Finite Element Analysis Of Large Sparse Linear Matrix Equations

Development of computer technology promotes the development of numerical computational science and engineering science. To meet the need of the computer's high-speed and high-capacity, the new computational theory and computational methods was widely proposed. Because the Sparse linear equations are becoming larger and larger and the matrix is becoming more complex, the traditional methods can't solve the complex problem from engineering science. Therefore, due to both the development of computer technology and the need of solving complex engineering problems, the technology of solving linear equations must be changed.People often need solving the sparse linear equations from finite element analysis of electromagnetic field. The direct method of solving sparse linear equations can't make out the correct solutions. For overcoming the difficulty, the iteration methods is proposed, including Jacobi, Gauss-Siedel and SSor iteration. Subsequently, in order to solve the equations whose coefficient matrix is positive definite, the Krylov subspace method was discovered. The conjugate gradient method is one of the Krylov subspace method, it can solve the equations with less memory and high speed, it also can benefit from the sparse of the coefficient matrix. But the Krylov subspace method depend on the spectrum of the matrix, namely the eigenvalue of the matrix. So if the spectrum of the matrix is not focus, the conjugate method may converge slowly, even more can't converge. The preconditioner can solve this proplem, the preconditioner is any form of modification of an linear equation which can makes it easier to solve by a given iterative method.There are several kinds of iterative methods, including SAI Preconditioner, Diagonal Scaling Preconditioner, Incomplete LU Factorization Preconditioner. The Incomplete LU Factorization Preconditioner is derived by Gaussian elimination and dropping some elements with any kind of dropping strategy. This method is very widely used to solve the linear equations. The theory of solving equations including the preconditioner technology has been studied fully, and their performance is verified in the Matlab by many people. Some simulation software includes the modules which can solve linear equations. In order to solve the equations from finite element analysis of electromagnetic field, here are several software modules which are programmed with C/C++ in order to solve the sparse linear equation by Conjugate Gradient method, then modify the ILU++ package to solve linear equations ,by the Incomplete Preconditioner , Iteration .At last compare the performance of the package with the performance of GSS so as to determine whether the package can be used in the engineering .