改进的吸收边界条件和无条件稳定时域有限差分算法的研究

发布时间：2018-04-09 06:43

本文选题：时域有限差分法　切入点：完全匹配层　出处：《安徽大学》2017年硕士论文

【摘要】：时域有限差分法(Finite-Difference Time-Domain,FDTD)自提出以来在电磁计算领域得到了广泛的研究,并已经发展成为一种成熟的数值计算方法。在实际计算时,通常会遇到半开放和开放区域的问题,然而计算资源是有限的,因此吸收边界条件对于FDTD区域的截断至关重要。另外,FDTD算法的时间步长需满足Courant-Friedrich-Levy(CFL)稳定性条件,这使其在一些包含细小结构的仿真模型中的计算效率不高。基于上述问题,本文的研究内容是围绕吸收边界条件的吸收性能的改进以及无条件稳定的FDTD算法展开的。首先,本文简要介绍了一下FDTD算法的基础知识以及几种经典的完全匹配层(perfect match layer,PML)技术。接着,针对传统卷积完全匹配层(CPML)技术中存在的场值更新时刻不同步问题,提出了一种改进的CPML技术。与传统的CPML技术相比,这种改进的CPML的吸收性能更好,计算效率更高,并且没有增加CPML算法的复杂度。其次,对无条件稳定的交替方向隐式FDTD(Alternating-Direction-Explicit FDTD,ADI-FDTD)算法、弱条件稳定的混合显隐式FDTD(Hybrid Implicit-Explicit FDTD,HIE-FDTD)算法进行了介绍,详细推导了两种算法引入CPML吸收边界后的时域迭代公式,并给出了公式中场分量的迭代顺序,验证了CPML吸收边界在ADI-FDTD和HIE-FDTD算法中的正确性和有效性。最后,介绍了一种显式的无条件稳定算法即空间滤波FDTD(Spatially filtered FDTD,SF-FDTD)算法,与隐式的FDTD算法相比,它不需要复杂的公式推导及矩阵求逆运算,从而降低了无条件稳定算法在FDTD方法中应用的难度。传统SF-FDTD的不足之处在于只能应用在每个方向网格尺寸相同的区域中,基于此,本文提出了一种改进的SF-FDTD算法,使其能够应用在各方向网格尺寸不相同的区域中,扩大了 SF-FDTD算法的应用范围。接着,将SF-FDTD引入到亚网格技术中,由于SF-FDTD突破了 CFL条件的限制,在细网格区域可选取与粗网格区域相同的时间步长,有效地提高了计算效率。
[Abstract]:Finite-Difference Time-Domain FDTD (FDTD) has been widely studied in the field of electromagnetic computation since it was proposed, and has been developed into a mature numerical method.In practical computation, the problem of semi-open and open region is usually encountered. However, the computational resources are limited, so the absorbing boundary condition is very important for the truncation of the FDTD region.In addition, the time step size of FDTD algorithm needs to satisfy the stability condition of Courant-Friedrich-Levyn CFL, which makes it inefficient in some simulation models with small structures.Based on the above problems, this paper focuses on the improvement of the absorption performance of the absorbing boundary condition and the unconditionally stable FDTD algorithm.Firstly, this paper briefly introduces the basic knowledge of FDTD algorithm and several classical perfectly matched match layers.Then, an improved CPML technique is proposed to solve the problem of out-of-sync of field value update time in traditional convolution perfectly matched layer (CPML) technology.Compared with the traditional CPML technique, the improved CPML has better absorption performance and higher computational efficiency, and does not increase the complexity of the CPML algorithm.Secondly, the unconditionally stable alternating direction implicit FDTD(Alternating-Direction-Explicit FDTDU ADI-FDTD algorithm and the weakly conditional stable hybrid explicit implicit FDTD(Hybrid Implicit-Explicit FDTD- HIE-FDTD algorithm are introduced. The time domain iterative formulas of the two algorithms after introducing the CPML absorbing boundary are derived in detail.The iterative order of the field component of the formula is given to verify the correctness and validity of the CPML absorbing boundary in the ADI-FDTD and HIE-FDTD algorithms.Finally, an explicit unconditionally stable algorithm, spatial filtering FDTD(Spatially filtered FDTDX SF-FDTD algorithm, is introduced. Compared with the implicit FDTD algorithm, it does not require complicated formula derivation and matrix inversion.Therefore, the application difficulty of unconditional stability algorithm in FDTD method is reduced.The disadvantage of traditional SF-FDTD is that it can only be used in the same area of grid size in every direction. Based on this, an improved SF-FDTD algorithm is proposed to make it applicable to different areas with different grid sizes in different directions.The application of SF-FDTD algorithm is expanded.Then, SF-FDTD is introduced into subgrid technology. Because SF-FDTD breaks through the restriction of CFL condition, the time step of fine grid region can be chosen the same as that of coarse grid area, and the computational efficiency is improved effectively.
【学位授予单位】：安徽大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O441

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