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Reconstruction methods in inverse boundary value problems

逆边值问题的重构方法

基本信息

批准号：
13640115
负责人：
TANUMA Kazumi
金额：
$ 2.56万
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：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640115/
关键词：
inverse boundary value problems impedance tomography reconstruction formula crystallographic texture surface impedance tensor Rayleigh wave initial stress acoustoelastic coefficient texture 音弾性 Strohの定式化 Dirichlet to Neumann map 逆問

项目摘要

1.We considered the problem of determining conductivity of the medium from the measurements of the electric potential on the boundary and the corresponding current flux across the boundary, that is, from the Dirichlet to Neumann map. Three kinds of formulas for reconstructing conductivity and its normal derivative from the localized Dirichlet to Neumann map were obtained. They are the formulas for pointweise reconstruction, reconstruction in a weak form, and reconstruction in the form of Fourier transform. In these formulas, the normal derivative of the conductivity at the boundary is reconstructed from the localized Dirichlet to Neumann map without any information about the conductivity itself, which will give an efficient method in their numerical realization. Since our reconstruction formulas involve limiting process, we also gave the estimates for their convergences.2.The surface impedance tensor is a Hermitian second-order tensor which, for a homogeneous elastic half-space, maps the displacements given at the surface to the tractions needed to sustain them. We accounted for the effects of crystallographic texture in orthorhombic aggregates of cubic crystallites only up to terms linear in the texture coefficients and gave an explicit formula for the terms in the surface impedance tensor up to those linear in the texture coefficients. By the same method, we gave a dispersion formula for the velocity of Rayleigh waves propagating textured polycrystals with initial stress, which shows how the initial stress and the crystallographic texture have an influence, within terms linear in them, on the angular dependency of the Rayleigh waves velocity. This formula includes terms that describe the effects of texture on the acoustoelastic coefficients.

1.研究了由边界上的电势和相应的电流通量确定介质电导率的问题，即由Dirichlet到Neumann映射确定介质电导率的问题。得到了由定域Dirichlet到Neumann映射重建电导率及其法向导数的三种公式。它们分别是点间重建公式、弱形式重建公式和傅里叶变换形式重建公式。在这些公式中，电导率在边界处的法向导数是由局部化的Dirichlet到Neumann映射重构的，而不需要关于电导率本身的任何信息，这将为它们的数值实现提供一种有效的方法。由于我们的重建公式涉及到极限过程，我们也给出了它们的收敛性估计。2.对于均匀弹性半空间，表面阻抗张量是一个Hermitian二阶张量，它将表面上给定的位移映射到维持它们所需的力。我们占晶体织构的立方晶体的正交聚集体的影响，只有到条款线性的纹理系数，并给出了一个明确的公式中的条款的表面阻抗张量的线性纹理系数。用同样的方法，我们给出了瑞利波在有初应力的织构多晶体中传播的速度的色散公式，它表明初应力和晶体织构对瑞利波速度的角度依赖性有怎样的影响，在线性项内。该公式包括描述纹理对声弹性系数的影响的项。