Shape Identification of Defects in Structures Using the Adjoint Method

The inverse problem of accurately identifying obstacles in wave-supporting media is considered. We first solve the problem of accurately identifying the shape, size and location of a hole in a membrane which satisfies the scalar wave equation, and later extend it to the problem of accurately identifying an inclusion1 in an elastic medium. Both problems are extremely important, and appear in various fields such as structural damage identification, medical imaging and geophysical exploration. While most papers tackle these problems indirectly by constructing a spatial function that expresses the location and shape of the obstacle through changes in its values (e.g. Arrival Time Imaging Method, Time Reversal Method, Material Full Waveform Inversion method, etc.), we suggest here a direct, simple and intuitive method. We define an objective function which in both problems depends on the obstacle’s (hole or inclusion) boundary curve, and in the second problem depends also on the inclusion’s material properties. Then, a gradient based minimization algorithm, combined with the adjoint-method for its efficient calculation of the gradient, is applied. The focus in this work is on defining a completely general parametric representation for the unknown curve, on deriving an adjoint-method for it, and on testing the entire minimization process through a variety of numerical experiments. Many other aspects, such as meshing techniques, favorable conditions for identification, regularization, effect of noise, and more, are being discussed in detail.