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.
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.