探地雷达正演模拟的高效无网格数值计算方法研究

A high efficient field simulation algorithm determines the adaptability and stability of an inversion algorithm. It is very difficult to discretize a GPR model into meshes/grids since it has a multiscale structure and it is composed by a lot of complex materials. In contrast, meshless numerical method only bases on a set of scattering nodes which does not need to form a mesh and can be located arbitrarily to describe the geometries of interest. Therefore, meshless method has a strong competitive advantage in dealing with the multiscale problems. However, the formulas created by pure meshless approaches are generally identified as ill-posed and the solutions are prone to be unstable. To overcome the above problems, an unconditional stable radial point interpolation method (RPIM) is proposed. In this project, firstly, all of the nodes are arranged reasonable to avoid the interpolation matrix to become singular. Then, the eigenmodes of the matrix are analyzed to remove the unstable modes, which can fundamentally eliminate the solution instabilities and accelerate the meshless programs. Furthermore, the proposed schemes such as the nodes’ distribution, non-uniform support domain, and the hybrid basic functions are very easier to achieve since they are just dependent on the density of the nodes. It is unnecessary to introduce extra equations. The size of the support domain and the forms of the basic functions can be automatically changed. Therefore, the proposed method is simple and can be used in other computations. Based on the numerical model, the electromagnetic backscattering characteristics of the objects under complex soil intend to be researched, to provide the data and the theoretical basis for the GPR’s quantitative and automatic interpretation analysis.

正演模型的计算速度与精度决定了反演算法的适应性与稳定性。复杂探地雷达正演模型的多尺度结构为传统基于网格的数值方法的单元剖分造成困难，无网格数值方法仅基于节点运算，无需单元剖分，节点需求量少且分布灵活，在处理多尺度结构问题方面极具优势。然而，纯无网格法得到的方程组一般是病态的，其解是不稳定的。本申请项目以径向基点插值法（RPIM）为核心，提出一种无条件稳定的纯无网格数值算法。该方法首先合理安排节点，避免插值矩阵奇异，然后基于矩阵特征模分析，滤除导致算法不稳定的因素，从根本上实现算法的稳定与加速。项目中提出的节点多尺度分布方案、非均匀支撑域与多种基函数混合方案，只根据节点密度自动改变支撑域大小或基函数形式，无需引入额外的求解方程，方法简单且具普适性。基于数值模型，本项目拟对复杂土壤背景下的地下目标的回波响应特征展开研究，为实现探地雷达数据的定量解译与自动判读提供理论分析的方法和依据。

复杂探地雷达正演模型的多尺度结构为传统基于网格的数值方法的单元剖分造成困难，无网格数值方法仅基于节点运算，无需单元剖分，节点需求量少且分布灵活，在处理多尺度结构问题方面极具优势。然而，纯无网格法得到的方程组一般是病态的，其解是不稳定的。本项目以径向基点插值法（RPIM）为核心，主要对以下内容进行了研究：. 1）设计了三款能应用于UWB-GPR的平面印刷天线，为今后开展GPR系统的真实天线模拟提供技术支持；. 2）基于RPIM的有限节点环境对RPIM的数值色散情况进行分析，得到有耗空间的普适性结论；. 3）对基于矢量波动方程的RPIM算法进行研究，新算法在减少未知数的同时还能抑制非物理电荷的积累；. 4）对RPIM的稳定性问题进行分析、处理。首先从建立多尺度节点分布模型的角度出发提高算法的稳定度；然后通过分析RPIM的系数矩阵特征，引入截断奇异值分解（TSVD）法，采用系数矩阵的伪逆代替原始矩阵的逆运算，从而避免因矩阵无法求逆而导致的算法无法进一步运行的问题；接着，将Gaussian基函数使用Mercer原理展开为Gaussian核，寻找一个能够消除算法对形参依赖的新的基函数，并采用求解伪逆的方式代替矩阵求逆。现有的数值实验证明：以上处理方式能综合解决算法的精度与稳定性的问题。. 5）对基于差分法的子网格数值计算方法、无条件稳定显式迭代算法进行研究，并基于传统FDTD法、子网格FDTD法、无条件稳定的显式方法、RPIM法建立GPR数值计算模型。为今后开展无网格法与网格法间的混合建模提供必要的技术支持；. 6）以上算法被应用于天线测试、多物理场仿真、电磁波传播、腔体谐振、探地雷达回波信号模拟等电磁问题中，计算性能得到了有效证明。