基于PML边界条件的二阶电磁波动方程GPR时域有限元模拟
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摘要
完全匹配层(PML)作为一种稳定高效的吸收边界条件,广泛应用于基于一阶电磁波动方程的探地雷达(GPR)数值模拟中.为解决基于二阶电磁波动方程的GPR数值模拟的吸收边界问题,本文借鉴二阶弹性波动方程的PML边界条件构建思想,提出了一种适合二阶电磁波动方程GPR时域有限元模拟的PML边界条件.从二阶电磁波动方程出发,基于复拉伸坐标变换,推导了PML算法的频域表达式;通过合理构造辅助微分方程,得到了PML算法的时域表达式,并以变分形式(弱形式)加载到GPR时域有限元方程中,实现了PML边界条件在二阶电磁波动方程GPR时域有限元模拟中的应用.在此基础上,对比了无边界条件、Sarma边界条件和PML边界条件下均匀模型的波场快照、单道波形、时域反射误差和能量衰减曲线,结果表明:PML边界条件的吸收效果要远优于Sarma边界条件,具有近似零反射系数.一个复杂介质模型的正演模拟验证了PML边界条件在非均匀地电结构中电磁波传播模拟的良好吸收效果.
The perfect matching layer(PML),as a stable and efficient absorbing boundary condition,is widely used in the Ground Penetrating Radar(GPR)numerical simulation of firstorder electromagnetic wave equation.In order to solve the problem of the absorbing boundary in GPR numerical simulation based on second-order electromagnetic wave equation,this paper proposes a PML boundary condition for Finite Element Time Domain(FETD)simulation of GPR based on second order electromagnetic wave equation according to the PML boundary condition construction idea of second-order elastic wave equation.Taking the two-dimensional TM waveequation as an example,the frequency domain formula of PML algorithm is deduced according to the complex coordinate transformation and its time domain equation is obtained by constructing reasonable auxiliary differential equation.And the PML boundary condition is loaded into the FETD equation of GPR in a form of variation principle(weak form),so that the application of PML in the FETD simulation of GPR second-order electromagnetic wave equation is realized.On this basis,comparison of the wave field snapshots,signal waveforms,time reflection errors and energy attenuation curves of the homogenous model with Sarma,PML boundary condition and without this boundary condition demonstrates that the absorption effect of PML is much better than the Sarma boundary condition with a near-zero reflection coefficient.At last,the numerical simulation of a complex model verifies the good absorbing effect of the PML boundary condition on electromagnetic wave propagation in a heterogeneous geo-electric structure.
The perfect matching layer(PML),as a stable and efficient absorbing boundary condition,is widely used in the Ground Penetrating Radar(GPR)numerical simulation of firstorder electromagnetic wave equation.In order to solve the problem of the absorbing boundary in GPR numerical simulation based on second-order electromagnetic wave equation,this paper proposes a PML boundary condition for Finite Element Time Domain(FETD)simulation of GPR based on second order electromagnetic wave equation according to the PML boundary condition construction idea of second-order elastic wave equation.Taking the two-dimensional TM waveequation as an example,the frequency domain formula of PML algorithm is deduced according to the complex coordinate transformation and its time domain equation is obtained by constructing reasonable auxiliary differential equation.And the PML boundary condition is loaded into the FETD equation of GPR in a form of variation principle(weak form),so that the application of PML in the FETD simulation of GPR second-order electromagnetic wave equation is realized.On this basis,comparison of the wave field snapshots,signal waveforms,time reflection errors and energy attenuation curves of the homogenous model with Sarma,PML boundary condition and without this boundary condition demonstrates that the absorption effect of PML is much better than the Sarma boundary condition with a near-zero reflection coefficient.At last,the numerical simulation of a complex model verifies the good absorbing effect of the PML boundary condition on electromagnetic wave propagation in a heterogeneous geo-electric structure.
引文
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Feng D S,Dai Q W.2011.GPR numerical simulation of full wave field based on UPML boundary condition of ADI-FDTD.NDT&E International,44(6):495-504,doi:10.1016/j.ndteint.2011.05.001.
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Feng D S,Yang L Y,Wang X.2016b.The unsplit convolutional perfectly matched layer absorption performance analysis of evanescent wave in GPR FDTD forward modeling.Chinese Journal of Geophysics(in Chinese),59(12):4733-4746,doi:10.6038/cjg20161232.
Feng D S,Wang X.2017.Convolution perfectly matched layer for the finite-element time-domain method modeling of Ground Penetrating Radar.Chinese Journal of Geophysics(in Chinese),60(1):413-423,doi:10.6038/cjg20170134.
Feng D S,Wang X,Zhang B.2018.Specific evaluation of tunnel lining multi-defects by all-refined GPR simulation method using hybrid algorithm of FETD and FDTD.Construction and Building Materials,185:220-229,doi:10.1016/j.conbuildmat.2018.07.039.
Feng X,Zou L L,Liu C,et al.2011.Forward modeling for fullpolarimetric ground penetrating radar.Chinese Journal of Geophysics(in Chinese),54(2):349-357,doi:10.3969/j.issn.0001-5733.201.02.011.
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Giannopoulos A.2005.Modelling ground penetrating radar by GprMax.Construction and Building Materials,19(10):755-762,doi:10.1016/j.conbuildmat.2005.06.007.
Giannopoulos A.2008.An improved new implementation of complex frequency shifted PML for the FDTD method.IEEETransactions onAntennasandPropagation,56(9):2995-3000,doi:10.1109/TAP.2008.928789.
Hu Y P,Egbert G,Ji Y J,et al.2017.A novel CFS-PML boundary condition for transient electromagnetic simulation using a fictitious wave domain method.Radio Science,52(1):118-131,doi:10.1002/2016RS006160.
Irving J,Knight R.2006.Numerical modeling of ground-penetrating radar in 2-D using MATLAB.Computers and Geosciences,32(9):1247-1258,doi:10.1016/j.cageo.2005.11.006.
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