A direct linear inversion for discontinuous elastic parameters recovery from internal displacement information only

The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without...

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Bibliographic Details
Published in:Numerische Mathematik 2021, Vol.147 (1), p.189-226
Main Authors: Ammari, Habib, Bretin, Elie, Millien, Pierre, Seppecher, Laurent
Format: Article
Language:English
Subjects:
Analysis of PDEs
Bioengineering
Elastic recovery
Functional Analysis
Imaging
Inverse problems
Life Sciences
Linear operators
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical and Computational Physics
Parameters
Reconstruction
Shear modulus
Simulation
Smoothness
Stability
Stiffness
Tensors
Theoretical
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Summary:The aim of this paper is to present and analyze a new direct method for solving the linear elasticity inverse problem. Given measurements of some displacement fields inside a medium, we show that a stable reconstruction of elastic parameters is possible, even for discontinuous parameters and without boundary information. We provide a general approach based on the weak definition of the stiffness-to-force operator which conduces to see the problem as a linear system. We prove that in the case of shear modulus reconstruction, we have an L 2 stability with only one measurement under minimal smoothness assumptions. This stability result is obtained through the proof that the linear operator to invert has closed range. We then describe a direct discretization which provides stable reconstructions of both isotropic and anisotropic stiffness tensors.
ISSN:0029-599X
0945-3245
DOI:10.1007/s00211-020-01164-6