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。我们认为瑞利波沿着同质，各向异性，预应力的半空间的自由表面传播。我们假设预应性各向异性培养基偏离了由初始应力和递增弹性张量的各向异性部分正式引起的相对不受干扰，无压力和各向同性状态的比较。然而，没有假设对增量弹性张量的材料各向异性。在Stroh形式主义的帮助下，我们提出了一个雷利波的相位速度从其比较的各向同性值转移的一阶扰动公式。该公式明确显示了初始应力和各向异性部分如何影响瑞利波的相位速度。通过类似的参数，我们研究了极化比的扰动，这是位移的纵向分量的最大值与正常分量的最大值和相移的最大比率，这是相位从纵向分量到表面位移正常分量的相位的变化。我们还讨论了通过测量雷利波扰动的测量来确定初始应力和材料各向异性的问题。2。我们提供的公式将电导率及其正常衍生物从局部dirichlet到neumann映射的边界上重建。还提出了重建公式的数值实现3。我们考虑一个逆问题，可以通过观察单个初始数据产生的冲击来确定进入标量保护定律的通量功能。我们证明，间隔的通量函数可以由冲击唯一确定。我们还证明，可以通过选择适当的初始数据序列来任意将此间隔占用。