Conditional Stability for the Hexagonal Anisotropic Elastic Dynamical Systems

Lin, Chia Feng; Nakamura, Gen

doi:10.1080/03605300903111414

Cited by 2 publications

(4 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…As the proof of Theorem 2 relies on bounding the term E δ , it requires to estimate either the strain tensor ε n,δ in the L ∞ -norm and the hessian tensor H k n,δ in the L 2 -norm, or conversely. The choice adopted in (26) is actually the more practical.…”

Section: Solution Stability Wrt Noisy Displacement Measurementsmentioning

confidence: 99%

“…For simplicity, we assume that the function η(δ) in ( 26) is given by a power law. Now, the aim is to tune the parameters h 1 and h 2 as functions of δ to obtain the best possible estimate in (26). For the sake of argument, let ≥ 0 ≥ 2 such that u n , u n,δ ∈ H (Ω) ∩ W −1,∞ (Ω) for all δ > 0 and n = 1, 2.…”

Section: Construction Of Noisy Operator δ By Regularizationmentioning

confidence: 99%

“…The main tool for such proofs is the Runge approximation property, which itself follows from unique continuation property, established in e.g. [25][26][27] in the context of elasticity.…”

Section: Here We Globally Havementioning

confidence: 99%

See 2 more Smart Citations

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

et al. 2014

View full text Add to dashboard Cite

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach.

show abstract

Section: Solution Stability Wrt Noisy Displacement Measurementsmentioning

confidence: 99%

Section: Construction Of Noisy Operator δ By Regularizationmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

et al. 2014

View full text Add to dashboard Cite

show abstract

“…For the study of the inverse problems for the scalar partial differential equations with a finite number of observation, Bukhgeim and Klibanov [160] proposed a remarkable method based on a Carleman estimate. Later, the Carleman estimate method has been generalized to study the uniqueness and stability estimate of the solutions of the inverse problems for equations of elastodynamics by Isakov [208], Ikehata et al [203], Eller et al [182], Imanuvilov et al [204], Imanuvilov and Yamamoto [205,206], and Lin and Nakamura [241].…”

Section: Carleman Estimates and Uniqueness Of Solutions Formentioning

confidence: 99%

A Survey on Inverse Problems for Applied Sciences

Yaman

Yakhno²,

Potthast³

2013

Mathematical Problems in Engineering

View full text Add to dashboard Cite

The aim of this paper is to introduce inversion-based engineering applications and to investigate some of the important ones from mathematical point of view. To do this we employ acoustic, electromagnetic, and elastic waves for presenting different types of inverse problems. More specifically, we first study location, shape, and boundary parameter reconstruction algorithms for the inaccessible targets in acoustics. The inverse problems for the time-dependent differential equations of isotropic and anisotropic elasticity are reviewed in the following section of the paper. These problems were the objects of the study by many authors in the last several decades. The physical interpretations for almost all of these problems are given, and the geophysical applications for some of them are described. In our last section, an introduction with many links into the literature is given for modern algorithms which combine techniques from classical inverse problems with stochastic tools into ensemble methods both for data assimilation as well as for forecasting.

show abstract

Conditional Stability for the Hexagonal Anisotropic Elastic Dynamical Systems

Cited by 2 publications

References 18 publications

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

A Survey on Inverse Problems for Applied Sciences

Contact Info

Product

Resources

About