Asymptotic analysis for wave propagation in anisotropic elasticity and inverse problems

2018 Fiscal Year Final Research Report

• Project/Area Number: 26400157
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Mathematical analysis
• Research Institution: Gunma University
• Principal Investigator: Tanuma Kazumi 群馬大学, 大学院理工学府, 教授 (60217156)
• Research Collaborator: Man Chi-Sing ケンタッキー大学, 教授
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 非等方弾性体 / 弾性波動方程式 / 弾性表面波 / Rayleigh waves / 分散 dispersion / 摂動 perturbation / 残留応力 / 逆問題

Outline of Final Research Achievements

Elastic waves carry much information on the properties of the medium along which they propagate. In this research project we consider elastic surface waves (Rayleigh waves) which propagate along the surface of an anisotropic inhomogeneous elastic half-space. We give a perturbation formula for the velocity and for the polarization of the boundary displacements of Rayleigh waves, as caused by the anisotropy of the medium. We also give a high-frequency asymptotic formula for the velocity which expresses the frequency-dependence of the Rayleigh-wave velocity, i.e., the dispersion of Rayleigh waves, as caused by the inhomogeneity of the medium. These results allow us to investigate how anisotropy and inhomogeneity of the medium affect the velocity and the polarization of Rayleigh waves, and can be applied to the inverse problems of determining the anisotropy and the inhomogeneity of the medium when they are unknown from boundary measurement of velocities and polarizations of Rayleigh waves.

Free Research Field

応用解析学

Academic Significance and Societal Importance of the Research Achievements

本研究は，非等方性・非斉次性の同定（例として残留応力の非破壊評価）という多分野にわたる重要問題への，弾性波動方程式の数学解析によるアプローチであり，力学，数理物理，工学等の分野との接点を視野に入れた応用解析学からの貢献の一例である．弾性パラメターを決定する逆問題では，順問題解析のアルゴリズムをくり返し解くことで最適解を求める手法を適用するため，方程式のパラメターが解にどのように組み込まれていくかを追求する，精緻な順問題解析が必要となった．逆問題解析側の要請から順問題解析の発展が促される局面が提示できたことも，本研究の特色である．