基于变换光学有限差分探地雷达数值模拟研究

Study of GPR simulation based on the transformation optics FDTD

在采用有限差分方法开展探地雷达复杂目标体精细结构模拟时,为了提高计算精度,常采用非均匀网格对目标区域划分小尺寸的网格,以压制离散网格频散现象和保证有限差分方法的稳定性.常规非均匀网格和自适应亚网格技术在网格剖分数量和粗细网格边界处理上难以达到计算效率和计算精度的均衡.本文根据隐形斗篷(invisible cloak)理论,将基于变换光学(Transformation optics)理论应用于有限差分探地雷达数值计算中.该理论的主要思想是基于目标参数变化而保持电磁场的传播不变性,在坐标变换后,Maxwell方程的形式可以维持不变,而使得相对介电常数与磁导率的表达式变得复杂.通过这种方式可以虚拟地扩大目标体所占的网格节点数,减少背景介质区域的网格数,不增加模型空间的网格总数.另外,这种网格划分方式不但提高了计算效率,同时也可以克服亚网格技术边界反射误差的影响.本文推导实现了基于变换光学的二维有限差分方法,通过典型探地雷达模型测试,对比分析了该方法与常规有限差分、变网格有限差分和自适应亚网格有限差分的优缺点.计算结果验证了基于变换光学的有限差分可用于探地雷达目标精细结构模拟,具有较高的计算精度和计算效率.

In complex target fine structure GPR finite difference time domain（FDTD）simulation,in order to improve the computational accuracy,we commonly use the non-uniform grid to discretize small size grid in the target area to reduce the grid dispersion and keep stability.The conventional non-uniform grid and adaptive mesh refinement（AMR）sub-grid FDTD can not achieve balance between computational time and accuracy.In this paper,according to the invisible cloak theory,we apply the FDTD based on transformation optics（TO）in GPR complex target simulation.The basic theory of TO method is that the medium parameters change and the resulted electromagnetic field is invariant,and that the Maxwell equation can remain unchanged in coordinate transformation and make the expression of dielectric constant and magnetic permeability become more complex.It will increase the number of grid in target area and reduce that of background medium without changing the total grid.This grid mode can not only overcome the reflectionerror from sub-grid boundary,improve the target computational accuracy,but also save the computational time.This paper derives the basic formula of 2D FDFD with TO theory and compares the advantages and disadvantages of standard FDTD,Non-uniform FDTD,sub-grid FDTD and TO-FDTD with synthetic model.The results demonstrate that the TO-FDTD method can simulate the small target GPR signal response with high computational accuracy and efficiency.