2006 Fiscal Year Final Research Report Summary
Inverse problems for partial differential equations in mechanics and engineering science
| Project/Area Number |
16540095
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Gunma University |
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Principal Investigator |
TANUMA Kazumi Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (60217156)
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| Co-Investigator(Kenkyū-buntansha) |
NAKAMURA Gen Hokkaido University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (50118535)
ASHINO Ryuichi Osaka Kyoiku University, Faculty of Education, Professor, 教育学部, 教授 (80249490)
IKEHATA Masaru Gunma University, Faculty of Engineering, Professor, 工学部, 教授 (90202910)
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| Project Period (FY) |
2004 – 2006
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| Keywords | clasticity equation / anisotropic elasticity / inverse problems / conservation law / Rayleigh wave / Stroh formation / impedance tomography / Dirichlet to Neumann map |
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| Research Abstract |
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.
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Research Products
(10 results)
