FDTD并行算法及层状半空间散射问题研究
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摘要
本文系统深入地研究了基于MPI的时域有限差分(Finite Difference Time Domain,FDTD)并行算法及层状半空间介质(包括色散和非色散介质)散射FDTD计算中入射波引入的关键技术。
基于MPI平台的FDTD并行算法主要包括:FDTD计算域的区域分解,子域间电磁场量的数据通信等;本文针对直角坐标系下并行FDTD的特点,创建了笛卡儿三维拓扑结构,采用捆绑式发送和接收的数据通信方式,优化了通信次序,克服了因通信次序造成的死锁问题;数据通信中,采用自定义的派生数据类型,避免了频繁的数据打包和解包过程,节省了打包和解包过程中中间存储变量占用的空间,提高了子域间数据通信速度,获得了较高的计算效率。
详细讨论了二阶Mur、UPML和CPML三种吸收边界的并行化处理方法,并分析了它们的负载均衡问题。同时,考察了三种吸收边界的性能差异,结果表明:以CPML为截断边界的三维并行FDTD算法,具有并行程序易于实现、吸收效果好和计算效率高等诸多优点,如CPML边界的并行FDTD对电大尺寸卫星模型的几个测试,效率都在90%以上。此外,还讨论了色散介质和各向异性介质的FDTD并行计算问题。采用并行算法对外形复杂(直升机和复合弹翼)和材质复杂(复合弹翼和复合各向异性介质球)目标的电磁散射特性进行了计算,数值计算结果表明本文算法和程序的有效性。
在层状半空间特别是色散、有耗层状半空间内目标散射问题的研究中,平面波源的引入是一个基本的、亟待克服的难题。本文详细论述了层状半空间问题FDTD计算中平面波引入的混合方式:垂直于分层界面的纵向侧边界上采用一维修正Maxwell方程(1D MME)计算分层空间斜入射平面波;自由空间内总场-散射场(TF/SF)上边界处的场分量采用“投影”插值而获得。为使TF/SF下边界处的单向行波形成无反射的吸收,将TF/SF下边界深入到完全匹配层内。这种TF/SF边界在二维面上形成的是Π型结构,区别于自由空间中的封闭结构。
为实现二维情形下层状半空间FDTD计算的平面波的引入,详细讨论了TM和TE两种模式的1D MME。在色散介质中,TE模式1D MME的离散处理相对TM模式比较困难,本文通过引入中间辅助变量,用辅助方程来解决这个问题。在离散辅助方程过程中采用三步平均技术,提高了计算精度和稳定性。把TE模式1D MME应用到Deybe、Lorentz和Drude三种常见的色散介质中,推导出了适用于三种介质类型的、中间变量到磁场变量的统一FDTD递推式。
在三维CPML吸收边界条件基础之上,分别导出了TM和TE模式的二维CPML吸收边界条件,以及适用于一维修正FDTD(1D-MFDTD)的一维CPML吸收边界条件,并成功地将一、二和三维CPML吸收边界条件应用于色散、有耗介质分层半空间问题FDTD计算的截断。
对1D-MFDTD计算的Courant稳定性分析表明,在入射角大于60°的斜入射情况,电磁波沿y方向相速度相对较大,降低了1D-MFDTD数值计算的稳定性。
本文用多个算例从不同方面考察色散层状半空间FDTD算法的可靠性,结果表明1D-MFDTD与Fourier方法得到的结果吻合。在层状半空间二维FDTD入射波源成功引入的基础上,实现了层状半空间三维FDTD计算中激励波源的引入。对平面波斜入射的二维和三维FDTD计算,形成完好的入射、反射和透射。提出并实现了适用于层状半空间电大尺寸目标的三维FDTD并行算法。数值模拟结果表明,该并行算法与串行算法的计算结果完全吻合。用若干实际算例分析了该算法的实际研究意义和价值。通过探雷算例验证了基于层状半空间并行FDTD算法和程序的高效性。
This dissertation lays a strong emphasis on the essential elements of a parallel algorithm for the Finite Difference Time Domain (FDTD) method based on MPI platform and the key techniques of plane wave injection in layered half-space to EM scattering analysis.
Based on the MPI (Message Passing Interface) library, an MPI Cartesian three-dimensional (3D) topology is used. A 3D Cartesian topology is defined and the FDTD computation domain is divided into some subspaces using a spatial decomposition of the regular grid structure. Then the fields inside each sub-domain are computed on an individual processor with a small amount of data being communicated from neighborhood sub-domain. For each field component, all the sub-domains send data in one direction sub-domain and receive data from opposite direction simultaneously. The transmission and reception can therefore be done with only one instruction, such as the procedure of SendRecv(), which optimizes the communication order and avoids the deadlock caused by this order. The inter-process communications are optimized by the use of derived data types, which group the data even when their memory addresses are not continuous, then the discontinuous data can be transmitted only once. Thus the optimized 3D parallel FDTD program can be resulted.
This dissertation discusses with emphasis on the parallel processing method to the absorbing boundary condition (ABC) including the second-order Mur`s absorbing condition, the uniaxial anisotropic perfectly matched layer (UPML) and convolutional perfectly matched layer (CPML). The load balance and absorbing performance of these ABCs are also analyzed. The CPML ABC has the advantages of high performance in parallel computation, readily programming and high-absorbing as well. The efficiency up to 90% calculated by parallel FDTD with CPML for large-size satellite is reached. It applies not only to the isotropic but also to the dispersive and the anisotropic media. The numerical results to several dispersive models validate our parallel algorithm. The radar cross sections (RCS) to the targets of complex shape, such as helicopter and compound wing, and the targets of complex material, such as the compound isotropic media wing and compound anisotropic media sphere are analyzed by using the parallel FDTD method.
In application of FDTD to the scattering analysis of object embedded in layered half-space, especially to the dispersive and lossy media, the injection of incident electromagnetic plane wave becomes complicated, because the traditional method in free space is not applicable. The plane wave to 3D FDTD in layered half-space is in fact a 2D problem. To solve this problem, the hybrid scheme is presented: the incident wave along side TF-SF boundaries are governed by the 1D modified Maxwell`s equations (1D MME); the incident wave on the upper TF-SF located in free space is in fact a duplication of the waveform at the cross points of TF-SF boundary with a proper time delay; the lower TF-SF boundary can also be treated by the same way as the upper, if the lowest layer medium is non-dispersive and lossless. However, the lowest layer medium with dispersive and lossy medium is of interested, and then we extend the side TF-SF downwards into CPML, which makes sure that the downward traveling wave is absorbed at lower media by the ABC. The presented hybrid scheme withΠ-shape being unclosed, it is unnecessary to determine the incidence wave along the lower TF-SF as it does in the traditional closed scheme.
To inject the plane wave into layer half-space in 2D FDTD, the formulations to 1D MME for TM and TE mode are developed. Furthermore, the discretization to TE mode is more difficult. Then an auxiliary magnetic variable is used, the 1D modified FDTD (MFDTD) to TE mode without any approximation and a three-step averaging technique in discretization to ensure convergence and improve precision is proposed. Considering three classes of dispersive material of Debye, Lorentz and Drude media, a uniform auxiliary update equation is derived.
Based on 3D CPML ABC, the CPML ABCs applicable to 2D and 1D case for TM and TE modes are derived, respectively. Then, these ABCs are applied to truncating FDTD domain in dispersive and lossy media successfully.
The Courant stability criterion in 1D-MFDTD is analyzed for different incident angles. The result shows that 1D-MFDTD is of instability while the incident angle is greater than 60°, because the phase velocity becomes larger along the y-direction . The feasibility of the algorithm is validated by using several dispersive layered half-space models from different aspects. The results by 1D MFDTD are in good agreement with Fourier transform method. The plane wave is introduced in 3D FDTD using the result of 2D FDTD computation, 2D and 3D FDTD simulations produce perfectly plane wave consisting of incident, reflected and transmitted wave. The parallel algorithm is also proposed and implemented to deal with the large-size target scattering in layered half-space. The numerical results show that the parallel and serial algorithms can obtain the same result. The significance and valuable application of this study are proved by several practical models. Finally, the model of land mine detection indicates the high performance to the parallel FDTD algorithm in layered half-space.
基于MPI平台的FDTD并行算法主要包括:FDTD计算域的区域分解,子域间电磁场量的数据通信等;本文针对直角坐标系下并行FDTD的特点,创建了笛卡儿三维拓扑结构,采用捆绑式发送和接收的数据通信方式,优化了通信次序,克服了因通信次序造成的死锁问题;数据通信中,采用自定义的派生数据类型,避免了频繁的数据打包和解包过程,节省了打包和解包过程中中间存储变量占用的空间,提高了子域间数据通信速度,获得了较高的计算效率。
详细讨论了二阶Mur、UPML和CPML三种吸收边界的并行化处理方法,并分析了它们的负载均衡问题。同时,考察了三种吸收边界的性能差异,结果表明:以CPML为截断边界的三维并行FDTD算法,具有并行程序易于实现、吸收效果好和计算效率高等诸多优点,如CPML边界的并行FDTD对电大尺寸卫星模型的几个测试,效率都在90%以上。此外,还讨论了色散介质和各向异性介质的FDTD并行计算问题。采用并行算法对外形复杂(直升机和复合弹翼)和材质复杂(复合弹翼和复合各向异性介质球)目标的电磁散射特性进行了计算,数值计算结果表明本文算法和程序的有效性。
在层状半空间特别是色散、有耗层状半空间内目标散射问题的研究中,平面波源的引入是一个基本的、亟待克服的难题。本文详细论述了层状半空间问题FDTD计算中平面波引入的混合方式:垂直于分层界面的纵向侧边界上采用一维修正Maxwell方程(1D MME)计算分层空间斜入射平面波;自由空间内总场-散射场(TF/SF)上边界处的场分量采用“投影”插值而获得。为使TF/SF下边界处的单向行波形成无反射的吸收,将TF/SF下边界深入到完全匹配层内。这种TF/SF边界在二维面上形成的是Π型结构,区别于自由空间中的封闭结构。
为实现二维情形下层状半空间FDTD计算的平面波的引入,详细讨论了TM和TE两种模式的1D MME。在色散介质中,TE模式1D MME的离散处理相对TM模式比较困难,本文通过引入中间辅助变量,用辅助方程来解决这个问题。在离散辅助方程过程中采用三步平均技术,提高了计算精度和稳定性。把TE模式1D MME应用到Deybe、Lorentz和Drude三种常见的色散介质中,推导出了适用于三种介质类型的、中间变量到磁场变量的统一FDTD递推式。
在三维CPML吸收边界条件基础之上,分别导出了TM和TE模式的二维CPML吸收边界条件,以及适用于一维修正FDTD(1D-MFDTD)的一维CPML吸收边界条件,并成功地将一、二和三维CPML吸收边界条件应用于色散、有耗介质分层半空间问题FDTD计算的截断。
对1D-MFDTD计算的Courant稳定性分析表明,在入射角大于60°的斜入射情况,电磁波沿y方向相速度相对较大,降低了1D-MFDTD数值计算的稳定性。
本文用多个算例从不同方面考察色散层状半空间FDTD算法的可靠性,结果表明1D-MFDTD与Fourier方法得到的结果吻合。在层状半空间二维FDTD入射波源成功引入的基础上,实现了层状半空间三维FDTD计算中激励波源的引入。对平面波斜入射的二维和三维FDTD计算,形成完好的入射、反射和透射。提出并实现了适用于层状半空间电大尺寸目标的三维FDTD并行算法。数值模拟结果表明,该并行算法与串行算法的计算结果完全吻合。用若干实际算例分析了该算法的实际研究意义和价值。通过探雷算例验证了基于层状半空间并行FDTD算法和程序的高效性。
This dissertation lays a strong emphasis on the essential elements of a parallel algorithm for the Finite Difference Time Domain (FDTD) method based on MPI platform and the key techniques of plane wave injection in layered half-space to EM scattering analysis.
Based on the MPI (Message Passing Interface) library, an MPI Cartesian three-dimensional (3D) topology is used. A 3D Cartesian topology is defined and the FDTD computation domain is divided into some subspaces using a spatial decomposition of the regular grid structure. Then the fields inside each sub-domain are computed on an individual processor with a small amount of data being communicated from neighborhood sub-domain. For each field component, all the sub-domains send data in one direction sub-domain and receive data from opposite direction simultaneously. The transmission and reception can therefore be done with only one instruction, such as the procedure of SendRecv(), which optimizes the communication order and avoids the deadlock caused by this order. The inter-process communications are optimized by the use of derived data types, which group the data even when their memory addresses are not continuous, then the discontinuous data can be transmitted only once. Thus the optimized 3D parallel FDTD program can be resulted.
This dissertation discusses with emphasis on the parallel processing method to the absorbing boundary condition (ABC) including the second-order Mur`s absorbing condition, the uniaxial anisotropic perfectly matched layer (UPML) and convolutional perfectly matched layer (CPML). The load balance and absorbing performance of these ABCs are also analyzed. The CPML ABC has the advantages of high performance in parallel computation, readily programming and high-absorbing as well. The efficiency up to 90% calculated by parallel FDTD with CPML for large-size satellite is reached. It applies not only to the isotropic but also to the dispersive and the anisotropic media. The numerical results to several dispersive models validate our parallel algorithm. The radar cross sections (RCS) to the targets of complex shape, such as helicopter and compound wing, and the targets of complex material, such as the compound isotropic media wing and compound anisotropic media sphere are analyzed by using the parallel FDTD method.
In application of FDTD to the scattering analysis of object embedded in layered half-space, especially to the dispersive and lossy media, the injection of incident electromagnetic plane wave becomes complicated, because the traditional method in free space is not applicable. The plane wave to 3D FDTD in layered half-space is in fact a 2D problem. To solve this problem, the hybrid scheme is presented: the incident wave along side TF-SF boundaries are governed by the 1D modified Maxwell`s equations (1D MME); the incident wave on the upper TF-SF located in free space is in fact a duplication of the waveform at the cross points of TF-SF boundary with a proper time delay; the lower TF-SF boundary can also be treated by the same way as the upper, if the lowest layer medium is non-dispersive and lossless. However, the lowest layer medium with dispersive and lossy medium is of interested, and then we extend the side TF-SF downwards into CPML, which makes sure that the downward traveling wave is absorbed at lower media by the ABC. The presented hybrid scheme withΠ-shape being unclosed, it is unnecessary to determine the incidence wave along the lower TF-SF as it does in the traditional closed scheme.
To inject the plane wave into layer half-space in 2D FDTD, the formulations to 1D MME for TM and TE mode are developed. Furthermore, the discretization to TE mode is more difficult. Then an auxiliary magnetic variable is used, the 1D modified FDTD (MFDTD) to TE mode without any approximation and a three-step averaging technique in discretization to ensure convergence and improve precision is proposed. Considering three classes of dispersive material of Debye, Lorentz and Drude media, a uniform auxiliary update equation is derived.
Based on 3D CPML ABC, the CPML ABCs applicable to 2D and 1D case for TM and TE modes are derived, respectively. Then, these ABCs are applied to truncating FDTD domain in dispersive and lossy media successfully.
The Courant stability criterion in 1D-MFDTD is analyzed for different incident angles. The result shows that 1D-MFDTD is of instability while the incident angle is greater than 60°, because the phase velocity becomes larger along the y-direction . The feasibility of the algorithm is validated by using several dispersive layered half-space models from different aspects. The results by 1D MFDTD are in good agreement with Fourier transform method. The plane wave is introduced in 3D FDTD using the result of 2D FDTD computation, 2D and 3D FDTD simulations produce perfectly plane wave consisting of incident, reflected and transmitted wave. The parallel algorithm is also proposed and implemented to deal with the large-size target scattering in layered half-space. The numerical results show that the parallel and serial algorithms can obtain the same result. The significance and valuable application of this study are proved by several practical models. Finally, the model of land mine detection indicates the high performance to the parallel FDTD algorithm in layered half-space.
引文
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