An inverse problem for Voronoi diagrams: A simplified model of non-destructive testing with ultrasonic arrays
An inverse problem for Voronoi diagrams: A simplified model of non-destructive testing with ultrasonic arrays
复制标题
Voronoi 图的反问题：超声阵列无损检测的简化模型
DOI：
10.1002/mma.6977
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发表时间：
2020
影响因子：
2.9
通讯作者：
Bourne D
中科院分区：
文献类型：
--
作者：
Bourne D
在本文中，我们研究的逆问题，恢复的空间变化的材料特性的固体多晶物体从超声走时测量之间的点对躺在域边界。我们考虑一个恒定密度的介质，其中材料的晶格结构的取向以分段恒定的方式变化，产生局部各向异性区域，其中波速根据入射波方向和材料的已知慢度曲线而变化。这个特殊的问题受到当前多晶固体超声无损检测所面临的挑战的启发。我们使用Voronoi镶嵌模型的材料的几何形状和研究两个简化的反问题，我们忽略波折射。在第一个问题中，Voronoi几何本身和与每个区域相关联的方向是未知的。我们使用多起点非线性最小二乘法解决这个非光滑、非凸优化问题。实现了良好的重建，但该方法被证明是敏感的噪声添加。第二个问题考虑在固定的正方形网格上重建方向。这是一个平滑的优化问题，但有更多的自由度。我们证明，方向可以确定唯一的足够的边界测量，并提供了一个数值方法，是更稳定的噪音。
In this paper, we study the inverse problem of recovering the spatially varying material properties of a solid polycrystalline object from ultrasonic travel time measurements taken between pairs of points lying on the domain boundary. We consider a medium of constant density in which the orientation of the material's lattice structure varies in a piecewise constant manner, generating locally anisotropic regions in which the wave speed varies according to the incident wave direction and the material's known slowness curve. This particular problem is inspired by current challenges faced by the ultrasonic non‐destructive testing of polycrystalline solids. We model the geometry of the material using Voronoi tessellations and study two simplified inverse problems where we ignore wave refraction. In the first problem, the Voronoi geometry itself and the orientations associated to each region are unknowns. We solve this nonsmooth, nonconvex optimisation problem using a multistart non‐linear least squares method. Good reconstructions are achieved, but the method is shown to be sensitive to the addition of noise. The second problem considers the reconstruction of the orientations on a fixed square mesh. This is a smooth optimisation problem but with a much larger number of degrees of freedom. We prove that the orientations can be determined uniquely given enough boundary measurements and provide a numerical method that is more stable with respect to the addition of noise.