148 research outputs found
Reconstruction of generic anisotropic stiffness tensors from partial data around one polarization
We study inverse problems in anisotropic elasticity using tools from
algebraic geometry. The singularities of solutions to the elastic wave equation
in dimension with an anisotropic stiffness tensor have propagation
kinematics captured by so-called slowness surfaces, which are hypersurfaces in
the cotangent bundle of that turn out to be algebraic varieties.
Leveraging the algebraic geometry of families of slowness surfaces we show
that, for tensors in a dense open subset in the space of all stiffness tensors,
a small amount of data around one polarization in an individual slowness
surface uniquely determines the entire slowness surface and its stiffness
tensor. Such partial data arises naturally from geophysical measurements or
geometrized versions of seismic inverse problems. Additionally, we explain how
the reconstruction of the stiffness tensor can be carried out effectively,
using Gr\"obner bases. Our uniqueness results fail for very symmetric (e.g.,
fully isotropic) materials, evidencing the counterintuitive claim that inverse
problems in elasticity can become more tractable with increasing asymmetry.Comment: 39 pages, 4 figures. Computer Code included in the ancillary files
folde
Ultrasonic imaging in highly heterogeneous backgrounds
This work formally investigates the differential evolution indicators as a
tool for ultrasonic tracking of elastic transformation and fracturing in
randomly heterogeneous solids. Within the framework of periodic sensing, it is
assumed that the background contains (i) a multiply connected set of
viscoelastic, anisotropic, and piecewise homogeneous inclusions, and (ii) a
union of possibly disjoint fractures and pores. The support, material
properties, and interfacial condition of scatterers in (i) and (ii) are
unknown, while elastic constants of the matrix are provided. The domain
undergoes progressive variations of arbitrary chemo-mechanical origins such
that its geometric configuration and elastic properties at future times are
distinct. At every sensing step multi-modal incidents
are generated by a set of boundary excitations, and the resulting scattered
fields are captured over the observation surface. The test data are then used
to construct a sequence of wavefront densities by solving the spectral
scattering equation. The incident fields affiliated with distinct pairs of
obtained wavefronts are analyzed over the stationary and evolving scatterers
for a suit of geometric and elastic evolution scenarios entailing both
interfacial and volumetric transformations. The main theorem establishes the
invariance of pertinent incident fields at the loci of static fractures and
inclusions between a given pair of time steps, while certifying variation of
the same fields over the modified regions. These results furnish a basis for
theoretical justification of differential evolution indicators for imaging in
complex composites which, in turn, enable the exclusive tomography of evolution
in a background endowed with many unknown features
Ultrasonic imaging in highly heterogeneous backgrounds
This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time tâ—¦ contains (i) a multiply connected set ofviscoelastic, anisotropic, and piece-wise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties, and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step tâ—¦, t1, . . ., multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scatteringequation. The incident fields affiliated with distinct pairs of obtained wavefronts are analyzed over the stationary and evolving scatterers for a suit ofgeometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basis for theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features
Mathematics and Algorithms in Tomography
This is the eighth Oberwolfach conference on the mathematics of tomography. Modalities represented at the workshop included X-ray tomography, sonar, radar, seismic imaging, ultrasound, electron microscopy, impedance imaging, photoacoustic tomography, elastography, vector tomography, and texture analysis
Computational Multiscale Methods
Many physical processes in material sciences or geophysics are characterized by inherently complex interactions across a large range of non-separable scales in space and time. The resolution of all features on all scales in a computer simulation easily exceeds today's computing resources by multiple orders of magnitude. The observation and prediction of physical phenomena from multiscale models, hence, requires insightful numerical multiscale techniques to adaptively select relevant scales and effectively represent unresolved scales. This workshop enhanced the development of such methods and the mathematics behind them so that the reliable and efficient numerical simulation of some challenging multiscale problems eventually becomes feasible in high performance computing environments
- …