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Inverse problems for partial differential equations in mechanics and engineering science

Research Project

Project/Area Number 16540095
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionGunma University

Principal Investigator

TANUMA Kazumi  Gunma University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (60217156)

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)
Project Period (FY) 2004 – 2006
Project Status Completed (Fiscal Year 2006)
Budget Amount *help
¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 2004: ¥1,300,000 (Direct Cost: ¥1,300,000)
Keywordsclasticity 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・トモグラフィー / 数値シミュレーション / 流れ関数 / 一意性
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.

Report

(4 results)
  • 2006 Annual Research Report   Final Research Report Summary
  • 2005 Annual Research Report
  • 2004 Annual Research Report
  • Research Products

    (13 results)

All 2006 2005 2004 Other

All Journal Article (13 results)

  • [Journal Article] Perturbation formula for phase velocity of Rayleigh waves in prestressed anisotropic media2006

    • Author(s)
      Kazumi Tanuma
    • Journal Title

      Journal of Elasticity 85

      Pages: 21-37

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Perturbation of Rayleigh-waves velocity caused by a fully anisotropic term2006

    • Author(s)
      Kazumi Tanuma
    • Journal Title

      Hokkaido University of Technical Report Series in Mathematics 109

      Pages: 24-24

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Annual Research Report 2006 Final Research Report Summary
  • [Journal Article] Perturbation formula for phase velocity of Rayleigh waves in prestressed anisotropic media2006

    • Author(s)
      Kazumi Tamura, Chi-Sing Man
    • Journal Title

      Journal of Elasticity 85

      Pages: 21-37

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Perturbation of Rayleigh-wave velocity caused by a fully anisotropic term2006

    • Author(s)
      Kazumi Tamura
    • Journal Title

      Hokkaido University of Technical Report Series in Mathematics 24

      Pages: 24-24

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Inverse problems for scalar conservation laws2005

    • Author(s)
      Hyeonbae Kang
    • Journal Title

      Inverse Problems 21

      Pages: 1047-1059

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary 2005 Annual Research Report
  • [Journal Article] Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map2005

    • Author(s)
      Gen Nakamura
    • Journal Title

      Computing 75

      Pages: 197-213

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Inverse problems for scalar conservation laws2005

    • Author(s)
      Hyeonbae Kang, Kazumi Tamura
    • Journal Title

      Inverse Problems 21

      Pages: 1047-1059

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map2005

    • Author(s)
      Gen Nakamura, Samuli Siltanen, Kazumi Tamura, Shengzhang Wang
    • Journal Title

      Computing 75

      Pages: 197-213

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Numerical recovery of conductivity at the boundary from the localized Dirichlet to Neumann map2005

    • Author(s)
      G.Nakamura
    • Journal Title

      Computing (印刷中)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Inverse Problems for scalar conservation laws2004

    • Author(s)
      H.Kang
    • Journal Title

      Seminar Notes of Mathematical Sciences 7

      Pages: 50-59

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Stroh formalism and Rayleigh waves

    • Author(s)
      Kazumi Tanuma
    • Journal Title

      Journal of Elasticity (expository article) (印刷中)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stroh formalism and Rayleigh waves Inverse problems for scalar conservation laws

    • Author(s)
      Kazumi Tamura
    • Journal Title

      expository article in Journal of Elasticity Inverse Problems (In Press)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2006 Final Research Report Summary
  • [Journal Article] Stroh formalism and Rayleigh waves

    • Author(s)
      Kazumi Tanuma
    • Journal Title

      Journal of Elasticity(expository article) (To appear)

    • Related Report
      2006 Annual Research Report

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Published: 2004-04-01   Modified: 2016-04-21  

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