时域多分辨方法研究及其在电磁散射中的应用
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摘要
随着隐身技术、宽带和超宽带雷达技术的迅速发展，迫切需要开展雷达目标宽频带电磁散射特性的理论分析与研究。时域数值方法通过简单的时频变换就能得到目标宽频带范围内的信息，从而实现对物理量和物理现象更深刻、更直观的理解，受到了广泛关注。时域多分辨（Multiresolution Time-Domain, MRTD）方法作为一种新型的全波时域数值算法，具有良好的线性色散特性，可以在保持相对较小的相差情况下采用更低的空间采样率。其空间采样率在理论上可以达到奈奎斯特（Nyquist）采样极限，即每最短波长取两个采样点，因而可以极大地节省计算机资源，缩短计算时间，提高计算效率。尤其对于电大尺寸目标，MRTD方法的计算优势更为明显。
本文主要研究工作与贡献如下：
1.对基于Daubechies尺度函数的MRTD （Daubechies-MRTD）方法和基于双正交Cohen-Daubechies-Feauveau（CDF）尺度函数和小波函数的MRTD（CDF-MRTD）方法进行理论研究，详细推导了相应的电磁场计算的迭代公式。
2.详细分析了基于尺度函数的MRTD（Scaling MRTD, S-MRTD）方法的时间稳定性和空间数值色散特性。分析结果表明，MRTD方法的数值色散特性明显优于传统的时域有限差分（Finite-Differnce Time-Domain, FDTD）方法，但是MRTD方法的时间稳定性条件（Courant条件）要比传统FDTD方法苛刻，这说明MRTD方法是用时间换取空间。
3.研究了针对S-MRTD方法应用的连接边界条件。以Daubechies-S-MRTD方法为例，对二维TM极化和三维条件下的连接边界条件进行了详细地推导，首次给出了S-MRTD方法通用的完整的二维TM极化和三维条件下连接边界条件的“修正的迭代公式”。
4.研究了MRTD方法在电磁散射中的应用。包括各向异性完全匹配层（Anisotropic PerfectlyMatched Layer, APML）吸收边界条件、时谐场和瞬态场情况下的近场—远场外推方法等。进行了Daubechies-S-MRTD和CDF-S-MRTD方法电磁散射计算的数值试验。数值结果表明，本文研究的连接边界条件和APML吸收边界条件是有效的，而且这两种方法的计算效率均比传统FDTD方法更高。
5.根据Daubechies-S-MRTD方法的多区域分解技术，将Daubechies-S-MRTD方法与针对完全导电目标（Perfectly Electric Conductor, PEC）的局部共形FDTD（Conformal FDTD, CFDTD）算法结合，提出了一种针对PEC目标的基于Daubechies尺度函数的共形MRTD（Conformal MRTD,CMRTD）方法。PEC目标电磁散射计算的数值结果表明该方法能够有效降低Yee氏蛙跳式网格划分的台阶误差，明显提高计算的精度。该CMRTD方法也适用于CDF-S-MRTD方法。
6.基于局部共形技术和有效介电常数（Effective Dielectric Constant, EDC）概念，先后提出和研究了两种针对介质目标的基于Daubechies尺度函数的CMRTD方法，即尺度函数积分CMRTD（Scaling Functions Integral CMRTD, SFI-CMRTD）方法和多区域CMRTD（Multi-regionCMRTD, MR-CMRTD）方法。介质目标电磁散射计算的数值结果表明这两种方法均能有效解决麦克斯韦旋度方程在介质参数突变面处失效的问题以及Yee氏蛙跳式网格划分造成的台阶误差问题，可以明显提高计算精度。另外MR-CMRTD方法比SFI-CMRTD方法的计算效率更高，在分析电大尺寸介质目标时更具有优越性。
With the rapid development of stealth technology, wideband and ultra-wideband radar, thetheoretical analysis and research on wideband electromagnetic scattering characteristics of radar targetsare demanded urgently. By means of simple time-frequency transformation, the time-domain numericalmethods can obtain wideband information of targets and then achieve more profound and intuitionisticcomprehension about physical quantity and phenomenon, therefore have aroused great attention. As anew full-wave time-domain numerical method, the multiresolution time-domain (MRTD) scheme canapply lower sampling rate in space under the circumstance of remaining relatively less phase error dueto a good linear dispersion property. Its sampling rate can reach Nyquist sampling limit theoretically, i.e.two sampling points per shortest wavelength. So the MRTD scheme can hugely save computerresources, reduce computational time and then enhance computational efficiency. For electrically largetargets, especially, the MRTD scheme has more obvious advantage in computation.
The main researches and contributions of the thesis are summarized as follows:
1. The MRTD schemes based on Daubechies scaling functions (Daubechies-MRTD) andbiorthogonal Cohen-Daubechies-Feauveau scaling and wavelet functions (CDF-MRTD) are studiedtheoretically. The iterative equations of the electromagnetic fields are derived in detail.
2. The time stability and the space numerical dispersion property of the MRTD only based onscaling functions (S-MRTD) are dedailed analyzed. Analysis results show that the numerical dispersionproperties of the MRTD schemes are obvious better than those of the conventional finite-differncetime-domain (FDTD) method. But the time stability condition, i.e. Courant condition, of MRTDscheme is more rigorous than that of the conventional FDTD method, which explains that the MRTDscheme trades time for space in computing.
3. The connecting boundary condition aiming at the application of S-MRTD is studied. TakingDaubechies-S-MRTD as the example, the connecting boundary conditions under the two-dimensional(2D) TM polarized and three-demensional (3D) conditions are deduced in detail. And the complete“modified iterative equations” of the connecting boundary conditions under the2D TM polarized and3D conditions, which is general for all of the S-MRTD schems, are presented first.
4. The application of the MRTD scheme to electromagnetic scattering is investigated, whichincludes the anisotropic perfectly matched layer (APML) absorbing boundary condition andnear-to-far-field extrapolation method under the circumstances of time-harmonic and transient field, etc.
And the numerical tests of electromagnetic scattering computed by the Daubechies-S-MRTD andCDF-S-MRTD schemes are carried out. Numerical results show that the connecting boundary conditionand the APML absorbing boundary condition investigated in this thesis are effective, and thecomputational efficiency of the two schemes are better than that of the conventional FDTD method.
5. According to the multi-region decomposition technology of Daubechies-S-MRTD, a conformalMRTD (CMRTD) scheme based on Daubechies scaling functions used to the perfectly electricconductor (PEC) targets is proposed by combining the Daubechies-S-MRTD scheme with the locallyconformal FDTD (CFDTD) algorithm used to PEC targets. The numerical results of theelectromagnetic scattering computation of PEC targets demonstrate that the proposed CMRTD schemecan effectively reduce the staircase error of Yee’s leapfrog meshing and improve the computationalaccuracy obviously. The proposed CMRTD scheme can also be used in the CDF-S-MRTD scheme.
6. Based on locally conformal technology and the concept of effective dielectric constant (EDC),two CMRTD schemes based on Daubechies scaling functions used to dielectric targets, namely, thescaling functions integral CMRTD (SFI-CMRTD) and multi-region CMRTD (MR-CMRTD) scheme,are proposed and studied. The numerical results of the electromagnetic scattering computation ofdielectric targets show that both of the two schemes can solve the ineffectivity caused by thediscontinuous surface in dielectric case and the staircase error of Yee’s leapfrog meshing, and also canimprove computational accuracy obviously. Moreover, the MR-CMRTD scheme has bettercomputational efficiency than the SFI-CMRTD scheme, and has more advantage when analyzingelectrically large dielectric targets.
本文主要研究工作与贡献如下：
1.对基于Daubechies尺度函数的MRTD （Daubechies-MRTD）方法和基于双正交Cohen-Daubechies-Feauveau（CDF）尺度函数和小波函数的MRTD（CDF-MRTD）方法进行理论研究，详细推导了相应的电磁场计算的迭代公式。
2.详细分析了基于尺度函数的MRTD（Scaling MRTD, S-MRTD）方法的时间稳定性和空间数值色散特性。分析结果表明，MRTD方法的数值色散特性明显优于传统的时域有限差分（Finite-Differnce Time-Domain, FDTD）方法，但是MRTD方法的时间稳定性条件（Courant条件）要比传统FDTD方法苛刻，这说明MRTD方法是用时间换取空间。
3.研究了针对S-MRTD方法应用的连接边界条件。以Daubechies-S-MRTD方法为例，对二维TM极化和三维条件下的连接边界条件进行了详细地推导，首次给出了S-MRTD方法通用的完整的二维TM极化和三维条件下连接边界条件的“修正的迭代公式”。
4.研究了MRTD方法在电磁散射中的应用。包括各向异性完全匹配层（Anisotropic PerfectlyMatched Layer, APML）吸收边界条件、时谐场和瞬态场情况下的近场—远场外推方法等。进行了Daubechies-S-MRTD和CDF-S-MRTD方法电磁散射计算的数值试验。数值结果表明，本文研究的连接边界条件和APML吸收边界条件是有效的，而且这两种方法的计算效率均比传统FDTD方法更高。
5.根据Daubechies-S-MRTD方法的多区域分解技术，将Daubechies-S-MRTD方法与针对完全导电目标（Perfectly Electric Conductor, PEC）的局部共形FDTD（Conformal FDTD, CFDTD）算法结合，提出了一种针对PEC目标的基于Daubechies尺度函数的共形MRTD（Conformal MRTD,CMRTD）方法。PEC目标电磁散射计算的数值结果表明该方法能够有效降低Yee氏蛙跳式网格划分的台阶误差，明显提高计算的精度。该CMRTD方法也适用于CDF-S-MRTD方法。
6.基于局部共形技术和有效介电常数（Effective Dielectric Constant, EDC）概念，先后提出和研究了两种针对介质目标的基于Daubechies尺度函数的CMRTD方法，即尺度函数积分CMRTD（Scaling Functions Integral CMRTD, SFI-CMRTD）方法和多区域CMRTD（Multi-regionCMRTD, MR-CMRTD）方法。介质目标电磁散射计算的数值结果表明这两种方法均能有效解决麦克斯韦旋度方程在介质参数突变面处失效的问题以及Yee氏蛙跳式网格划分造成的台阶误差问题，可以明显提高计算精度。另外MR-CMRTD方法比SFI-CMRTD方法的计算效率更高，在分析电大尺寸介质目标时更具有优越性。
With the rapid development of stealth technology, wideband and ultra-wideband radar, thetheoretical analysis and research on wideband electromagnetic scattering characteristics of radar targetsare demanded urgently. By means of simple time-frequency transformation, the time-domain numericalmethods can obtain wideband information of targets and then achieve more profound and intuitionisticcomprehension about physical quantity and phenomenon, therefore have aroused great attention. As anew full-wave time-domain numerical method, the multiresolution time-domain (MRTD) scheme canapply lower sampling rate in space under the circumstance of remaining relatively less phase error dueto a good linear dispersion property. Its sampling rate can reach Nyquist sampling limit theoretically, i.e.two sampling points per shortest wavelength. So the MRTD scheme can hugely save computerresources, reduce computational time and then enhance computational efficiency. For electrically largetargets, especially, the MRTD scheme has more obvious advantage in computation.
The main researches and contributions of the thesis are summarized as follows:
1. The MRTD schemes based on Daubechies scaling functions (Daubechies-MRTD) andbiorthogonal Cohen-Daubechies-Feauveau scaling and wavelet functions (CDF-MRTD) are studiedtheoretically. The iterative equations of the electromagnetic fields are derived in detail.
2. The time stability and the space numerical dispersion property of the MRTD only based onscaling functions (S-MRTD) are dedailed analyzed. Analysis results show that the numerical dispersionproperties of the MRTD schemes are obvious better than those of the conventional finite-differncetime-domain (FDTD) method. But the time stability condition, i.e. Courant condition, of MRTDscheme is more rigorous than that of the conventional FDTD method, which explains that the MRTDscheme trades time for space in computing.
3. The connecting boundary condition aiming at the application of S-MRTD is studied. TakingDaubechies-S-MRTD as the example, the connecting boundary conditions under the two-dimensional(2D) TM polarized and three-demensional (3D) conditions are deduced in detail. And the complete“modified iterative equations” of the connecting boundary conditions under the2D TM polarized and3D conditions, which is general for all of the S-MRTD schems, are presented first.
4. The application of the MRTD scheme to electromagnetic scattering is investigated, whichincludes the anisotropic perfectly matched layer (APML) absorbing boundary condition andnear-to-far-field extrapolation method under the circumstances of time-harmonic and transient field, etc.
And the numerical tests of electromagnetic scattering computed by the Daubechies-S-MRTD andCDF-S-MRTD schemes are carried out. Numerical results show that the connecting boundary conditionand the APML absorbing boundary condition investigated in this thesis are effective, and thecomputational efficiency of the two schemes are better than that of the conventional FDTD method.
5. According to the multi-region decomposition technology of Daubechies-S-MRTD, a conformalMRTD (CMRTD) scheme based on Daubechies scaling functions used to the perfectly electricconductor (PEC) targets is proposed by combining the Daubechies-S-MRTD scheme with the locallyconformal FDTD (CFDTD) algorithm used to PEC targets. The numerical results of theelectromagnetic scattering computation of PEC targets demonstrate that the proposed CMRTD schemecan effectively reduce the staircase error of Yee’s leapfrog meshing and improve the computationalaccuracy obviously. The proposed CMRTD scheme can also be used in the CDF-S-MRTD scheme.
6. Based on locally conformal technology and the concept of effective dielectric constant (EDC),two CMRTD schemes based on Daubechies scaling functions used to dielectric targets, namely, thescaling functions integral CMRTD (SFI-CMRTD) and multi-region CMRTD (MR-CMRTD) scheme,are proposed and studied. The numerical results of the electromagnetic scattering computation ofdielectric targets show that both of the two schemes can solve the ineffectivity caused by thediscontinuous surface in dielectric case and the staircase error of Yee’s leapfrog meshing, and also canimprove computational accuracy obviously. Moreover, the MR-CMRTD scheme has bettercomputational efficiency than the SFI-CMRTD scheme, and has more advantage when analyzingelectrically large dielectric targets.
引文
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