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Inverse problems for partial differential equations in mechanics and engineering science

力学和工程科学中偏微分方程的反问题

基本信息

批准号：
16540095
负责人：
TANUMA Kazumi
金额：
$ 2.3万
依托单位：
Gunma University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540095/
关键词：
clasticity equation anisotropic elasticity inverse problems conservation law Rayleigh wave Stroh formation impedance tomography Dirichlet to Neumann map 弾性波動 polarization 弾性表面波非破壊検査 shock flux identification 導電体インピーダンス・トモグラ

项目摘要

1. We consider Rayleigh waves propagating along the free surface of a homogeneous, anisotropic, prestressed half-space. We assume that the deviation of the prestressed anisotropic medium from a comparative unperturbed, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we present a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. This formula shows explicitly how the initial stress and the anisotropic part, to first order of themselves, affect the phase velocity of Rayleigh waves. By the similar arguments we investigate the perturbation of the polarization ratio, which is the ratio of the maximum of the longitudinal component of the displacements to the maximum of the normal component, and of the phase shift, which is the shift in phase measured from that of the longitudinal component to that of the normal component of the displacements at the surface. We also discuss the problem of determining the initial stress and the material anisotropy by making measurements of perturbation of Rayleigh waves.2. We give formulae which reconstruct the conductivity and its normal derivative on the boundary of a planar disk domain from the localized Dirichlet to Neumann map. Numerical implementation of the reconstruction formulae is also presented.3. We consider an inverse problem to determine the flux function entering the scalar conservation law by observing the shock developed by a single initial data. We prove that the flux function on an interval can be uniquely determined by the shock. We also prove that this interval can be taken arbitrarily large by choosing an appropriate sequence of initial data.

1. 我们考虑沿均匀、各向异性、预应力半空间的自由表面传播的瑞利波。我们假设预应力各向异性介质与相对无扰动、无应力和各向同性状态的偏差很小，这在形式上是由初始应力和增量弹性张量的各向异性部分引起的。但未对增量弹性张量的材料各向异性作假设。利用Stroh形式，给出了瑞利波相速度相对各向同性值偏移的一阶摄动公式。该公式明确地显示了初始应力和各向异性部分对瑞利波相速度的影响。通过类似的论证，我们研究了极化比的扰动，极化比是位移的纵向分量的最大值与法向分量的最大值之比，以及相移的扰动，相移是从表面位移的纵向分量到法向分量测量的相移。本文还讨论了通过测量瑞利波的微扰来确定初始应力和材料各向异性的问题。从定域狄利克雷映射到诺伊曼映射，给出了在平面圆盘域边界上重构电导率及其法向导数的公式。给出了重构公式的数值实现。我们考虑了一个反问题，通过观察单个初始数据产生的激波来确定进入标量守恒律的通量函数。证明了区间上的通量函数可以由激波唯一确定。我们还证明了通过选择合适的初始数据序列可以取任意大的区间。