7 research outputs found
Reconstruction of a fully anisotropic elasticity tensor from knowledge of displacement fields
We present explicit reconstruction algorithms for fully anisotropic unknown
elasticity tensors from knowledge of a finite number of internal displacement
fields, with applications to transient elastography. Under certain
rank-maximality assumptions satified by the strain fields, explicit algebraic
reconstruction formulas are provided. A discussion ensues on how to fulfill
these assumptions, describing the range of validity of the approach. We also
show how the general method can be applied to more specific cases such as the
transversely isotropic one.Comment: 23 pages. Minor updates and additional reference
Imaging of isotropic and anisotropic conductivities from power densities in three dimensions
We present numerical reconstructions of anisotropic conductivity tensors in
three dimensions, from knowledge of a finite family of power density
functionals. Such a problem arises in the coupled-physics imaging modality
Ultrasound Modulated Electrical Impedance Tomography for instance. We improve
on the algorithms previously derived in [Bal et al, Inverse Probl Imaging
(2013), pp.353-375, Monard and Bal, Comm. PDE (2013), pp.1183-1207] for both
isotropic and anisotropic cases, and we address the well-known issue of
vanishing determinants in particular. The algorithm is implemented and we
provide numerical results that illustrate the improvements
Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method
We consider the nonlinear, inverse problem of identifying the stored energy
function of a hyperelastic material from full knowledge of the displacement
field as well as from surface sensor measurements. The displacement field is
represented as a solution of Cauchy's equation of motion, which is a nonlinear,
elastic wave equation. Hyperelasticity means that the first Piola-Kirchhoff
stress tensor is given as the gradient of the stored energy function. We assume
that a dictionary of suitable functions is available and the aim is to recover
the stored energy with respect to this dictionary. The considered inverse
problem is of vital interest for the development of structural health
monitoring systems which are constructed to detect defects in elastic materials
from boundary measurements of the displacement field, since the stored energy
encodes the mechanical peroperties of the underlying structure. In this article
we develope a numerical solver for both settings using the attenuated Landweber
method. We show that the parameter-to-solution map satisfies the local
tangential cone condition. This result can be used to prove local convergence
of the attenuated Landweber method in case that the full displacement field is
measured. In our numerical experiments we demonstrate how to construct an
appropriate dictionary and show that our algorithm is well suited to localize
damages in various situations.Comment: 28 pages, 6 figure
Applications of CGO Solutions on Coupled-Physics Inverse Problems
This paper surveys inverse problems arising in several coupled-physics
imaging modalities for both medical and geophysical purposes. These include
Photo-acoustic Tomography (PAT), Thermo-acoustic Tomography (TAT),
Electro-Seismic Conversion, Transient Elastrography (TE) and Acousto-Electric
Tomography (AET). These inverse problems typically consists of multiple inverse
steps, each of which corresponds to one of the wave propagations involved. The
review focus on those steps known as the inverse problems with internal data,
in which the complex geometrical optics (CGO) solutions to the underlying
equations turn out to be useful in showing the uniqueness and stability in
determining the desired information.Comment: 29 page
A foliated and reversible Finsler manifold is determined by its broken scattering relation
The broken scattering relation consists of the total lengths of broken
geodesics that start from the boundary, change direction once inside the
manifold, and propagate to the boundary. We show that if two reversible Finsler
manifolds satisfying a convex foliation condition have the same broken
scattering relation, then they are isometric. This implies that some
anisotropic material parameters of the Earth can be in principle reconstructed
from single scattering measurements at the surface
Inverse problem for compact Finsler manifolds with the boundary distance map
We prove that the boundary distance map of a smooth compact Finsler manifold
with smooth boundary determines its topological and differential structures. We
construct the optimal fiberwise open subset of its tangent bundle and show that
the boundary distance map determines the Finsler function in this set but not
in its exterior. If the Finsler function is fiberwise real analytic, it is
determined uniquely. We also discuss the smoothness of the distance function
between interior and boundary points
Stability for the inverse source problems in elastic and electromagnetic waves
This paper concerns the inverse source problems for the time-harmonic elastic
and electromagnetic wave equations. The goal is to determine the external force
and the electric current density from boundary measurements of the radiated
wave field, respectively. The problems are challenging due to the ill-posedness
and complex model systems. Uniqueness and stability are established for both of
the inverse source problems. Based on either continuous or discrete
multi-frequency data, a unified increasing stability theory is developed. The
stability estimates consist of two parts: the Lipschitz type data discrepancy
and the high frequency tail of the source functions. As the upper bound of
frequencies increases, the latter decreases and thus becomes negligible. The
increasing stability results reveal that ill-posedness of the inverse problems
can be overcome by using multi-frequency data. The method is based on integral
equations and analytical continuation, and requires the Dirichlet data only.
The analysis employs asymptotic expansions of Green's tensors and the
transparent boundary conditions by using the Dirichlet-to-Neumann maps. In
addition, for the first time, the stability is established on the inverse
source problems for both the Navier and Maxwell equations