Lester F.
Caudill, Jr.
The field of inverse problems
for physical systems is concerned with determining system properties which
cannot be measured by direct means. One measures other related quantities, and
seeks analytical and numerical methods for extracting the desired information
from these measurements. Nondestructive testing (NDT) is a subfield of inverse
problems in which one seeks to determine the interior properties of an object
by taking data measurements only on the surface of the object. My most recent
work focuses on two specific types of NDT -- thermal
imaging and inverse spectral problems in
vibration.
This technique is used to
determine the interior properties of an object by measuring their effect on
heat flow under controlled conditions. One uses a heat source (e.g. laser or
flashlamp) to apply a thermal flux to the surface of an object and then
observes the resulting temperature response on the object's surface. From this
information one attempts to determine internal thermal properties of the
object, for example, the presence of cracks or voids, or the shape of some
inaccessible portion of the surface. This process can be modeled as an inverse
boundary value problem for a partial differential equation. An important
potential application of the results of this work involves the detection and
characterization of corrosion in the interior of aircraft wings and fuselages.
[1]
K. Bryan and L.F. Caudill, Jr., An inverse problem in thermal imaging,
[2]
K. Bryan and L.F. Caudill, Jr., Stability and resolution in thermal
imaging, in Volume 3 of the Proceedings of the ASME Design Engineering
Technical Conferences, Boston, 1995, 1023-1032.
[3]
K. Bryan and L.F. Caudill, Jr., Uniqueness for a boundary identification
problem in thermal imaging, Electronic Journal of Differential Equations
C-1 (1997), 23-39.
[4]
K. Bryan and L.F. Caudill, Jr., Stability and reconstruction for an inverse
problem for the heat equation, Inverse Problems 14 (1998),
1429-1453.
[5]
K. Bryan and L.F. Caudill, Jr., Reconstruction of an unknown boundary portion from Cauchy data in n-dimensions, Inverse Problems 21 (2005), 239-255.
[6]
K. Bryan and L.F. Caudill, Jr., Algorithm-independent optimal input fluxes for boundary identification in thermal imaging, Journal of Physics Conference Series 124 (2008), 1-13.
The use of spectral methods in
NDT can be illustrated by considering the following experiment involving a beam
of unknown interior structure. Vibrations are induced in the beam, and the
resulting vibrational frequencies and normal mode shapes are measured. From
this data, one would like to infer something about the material properties
(e.g. density, bending stiffness, presence of structural damage) of the
interior of the beam. The vibrations are modeled by ordinary or partial
differential equations, and the vibrational frequencies enter as eigenvalues of
the resulting boundary value problems. Our goal, in joint work with Peter Perry
and Albert Schueller, is a thorough mathematical analysis of this inverse spectral
problem. One issue of particular interest is the identification of those
interior properties which can be determined by this process, and those which
are ``invisible" to it. One long-term goal of this project is to determine
the feasibility of vibrational testing methods as a means of damage assessment.
[1]
L.F. Caudill, Jr., P.A. Perry, and A.W. Schueller, Isospectral sets for
fourth-order ordinary differential operators,