On the Topological Derivative in Shape Optimization
SIAM Journal on Control and Optimization
The Topological Asymptotic for PDE Systems: The Elasticity Case
SIAM Journal on Control and Optimization
On derivative of energy functional for elastic bodies with cracks and unilateral conditions
Quarterly of Applied Mathematics
Imaging of location and geometry for extended targets using the response matrix
Journal of Computational Physics
SIAM Journal on Scientific Computing
Stability and Uniqueness for the Crack Identification Problem
SIAM Journal on Control and Optimization
Image Processing by Topological Asymptotic Expansion
Journal of Mathematical Imaging and Vision
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In this paper, we consider cracks with Dirichlet boundary conditions. We first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack. Based on this formula, we design a noniterative approach for locating a collection of small cracks. In order to do so, we construct a response matrix from the boundary measurements. The location and the length of the crack are estimated, respectively, from the projection onto the noise space and the first significant singular value of the response matrix. Indeed, the direction of the crack is estimated from the second singular vector. We then consider an extended crack with Dirichlet boundary conditions. We rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack. To reconstruct an extended crack from many boundary measurements, we develop two methods for obtaining a good guess. Several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave.