高效拉盖尔基时域有限差分算法研究

发布时间：2018-01-09 10:19

本文关键词：高效拉盖尔基时域有限差分算法研究　出处：《南京邮电大学》2014年博士论文　论文类型：学位论文

【摘要】：时域有限差分算法（FDTD）是目前应用最为广泛的电磁场数值计算方法之一，无条件稳定算法、计算效率和精度的提高以及吸收边界条件等课题一直是该算法研究的热点。拉盖尔基时域有限差分算法（WLP-FDTD）是一种以加权拉盖尔多项式作为时域基函数的FDTD法，此方法是无条件稳定的。该算法需要求解大型稀疏矩阵，对计算机内存需求大，计算效率低。近年来提出的高效WLP-FDTD算法，将大型稀疏矩阵转化成三对角矩阵，利用追赶法高效求解，减小了算法的内存需求，提高了计算效率。但是该算法由于引入了微扰项，产生了较大的计算误差。本文重点研究了新的高效WLP-FDTD算法，旨在提高WLP-FDTD算法的计算效率和计算精度。主要创新工作如下： 1.提出了一种新的二维高效WLP-FDTD算法。该算法在传统的WLP-FDTD算法中引入了新的微扰项，将大型稀疏矩阵方程转化成三对角方程求解。和现有的二维高效WLP-FDTD算法相比，减少了微扰项的项数，降低了微扰项中场微分的阶数，并且取消了两步算法中的非物理中间变量，，有效的减少了分裂误差，节省了计算机内存，提高了计算效率。在此基础上，又提出了基于该算法的迭代算法，进一步提高了算法的计算精度。 2.提出了一种新的三维高效WLP-FDTD算法。该算法是本文提出的新的二维高效WLP-FDTD算法向三维算法的推广，和现有的三维高效WLP-FDTD算法的区别在于引入的微扰项不同，可以有效的减少微扰项引入的分裂误差。算法中还引入了迭代算法，进一步提高了算法的计算精度。 3.从理论上证明了新的二维、三维高效WLP-FDTD迭代算法的收敛性；推导出了新的三维高效WLP-FDTD迭代算法的时域分段计算差分方程；并采用时域分段技术以谐振腔为例，验证了三维高效WLP-FDTD迭代算法的数值稳定性。 4.提出了适用于新的二维高效WLP-FDTD算法的PML、UPML吸收边界条件以及适用于新的三维高效WLP-FDTD算法的PML、CPML吸收边界条件，研究了匹配层参数对反射误差的影响规律。
[Abstract]:The finite difference time domain method (FDTD) is currently the most widely used electromagnetic field numerical calculation method of unconditionally stable algorithm, improve the computational efficiency and accuracy and absorbing boundary condition has been a hot topic in the research of algorithm. La Gail Ki finite-difference time-domain method (WLP-FDTD) is a kind of weighted Laguerre polynomials as FDTD method of time domain basis function, this method is unconditionally stable. This algorithm requires the solution of large sparse matrix, demand for computer memory and low computing efficiency. The efficient WLP-FDTD algorithm proposed in recent years, the transformation of a large sparse matrix into three diagonal matrix, by chasing method efficient solution, reduces the memory requirement however, the computational efficiency is improved. The algorithm due to the perturbation, produced a large calculation error. This paper focuses on the efficiency of the new WLP-FDTD algorithm, aiming at improving WLP-FDTD The calculation efficiency and accuracy of the algorithm. The main innovations are as follows:
1. this paper puts forward a new two-dimensional efficient WLP-FDTD algorithm. The algorithm in the traditional WLP-FDTD algorithm is introduced in the new perturbation, the large sparse matrix equation is transformed into solving three diagonal equations. Compared with two-dimensional, the existing WLP-FDTD algorithm, reduce the number of perturbation, the perturbation of low order reduction a midfielder differential, and the abolition of the non physical variables two step algorithm, effectively reduce the splitting errors, saving the computer memory and improve the computational efficiency. On this basis, and the iterative algorithm is proposed based on the algorithm, further improve the precision of the algorithm.
2. this paper puts forward a new efficient three-dimensional WLP-FDTD algorithm. The algorithm is a generalization of new efficient two-dimensional WLP-FDTD algorithm proposed in this paper to the three-dimensional algorithm, and the difference between the existing 3D WLP-FDTD algorithm is introduced, the different perturbation terms, can effectively reduce the perturbation into the split algorithm has introduced error. The iterative algorithm, to further improve the accuracy of the algorithm.
3. theoretically proved that the new two-dimensional, three-dimensional, convergence of the WLP-FDTD iterative algorithm is derived; subsection time domain WLP-FDTD iterative algorithm is a new efficient 3D calculation of the difference equation; and using time domain segmentation technology to the resonant cavity as an example, to verify the numerical stability of three-dimensional WLP-FDTD efficient iterative algorithm.
4., we proposed the PML, UPML absorbing boundary conditions suitable for the new two-dimensional efficient WLP-FDTD algorithm and the PML and CPML absorbing boundary conditions suitable for the new three-dimensional efficient WLP-FDTD algorithm. We also studied the influence of matching layer parameters on the reflection error.

【学位授予单位】：南京邮电大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TM15

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