4 INTRODUCTION Our method of asymptotic expansions of small-volume inclusions provides a use- ful framework to accurately and efficiently reconstruct the location and geometric features of the inclusions in a stable way, even for moderately noisy modal data. Indeed, the asymptotic expansions of the eigenvalue perturbations resulting from small perturbations of the shape of a conductivity inclusion, which extend those for small-volume conductivity inclusions, lead to very effective algorithms, aimed at determining certain properties of the shape of the conductivity inclusion based on eigenvalue measurements. We propose an original and promising opti- mization approach for reconstructing interface changes of a conductivity inclusion from measurements of eigenvalues and eigenfunctions associated with the transmis- sion problem for the Laplacian or the Lam´ e system. A key identity, dual to the asymptotic expansion for the perturbations in the modal measurements that are due to small changes in the interface of the inclusion, is established. It naturally leads to the formulation of the proposed optimization problem. The viability of our reconstruction algorithms is documented by a variety of numerical results. Their resolution limit is discussed. The case of multiple eigenvalues is rigorously handled as well. Our general approach is also applied for defect classification and sizing by vibration testing. Following the asymptotic formalism developed in this book, we derive asymptotic formulas for the effects of corrosion on resonance frequencies and mode shapes and use them to design a simple method for localizing the corrosion and estimating its extent. Our asymptotic theory for eigenvalue problems also leads to efficient algorithms for solving shape optimization problems. Shape optimization arises in many differ- ent fields, such as mechanical design and shape reconstruction. It can be generally described as a problem of finding the optimal shapes in a certain sense under certain constraints. We incorporate the asymptotic expansions derived in this book into a level set method to investigate optimal design of photonic and phononic crystals. The level set is used to represent the interface between two materials with different physical parameters. We present efficient algorithms for finding the optimal shapes for maximal band gaps and acoustic drum problems.
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