Optimized placement of distributed optical fiber sensors via shape derivatives: least-squares and mCRE approaches

We investigate the shape sensitivity of functionals arising in inverse problems involving distributed optical fiber sensors for structural health monitoring and material parameter identification. These sensors are deployed along curves, leading to objective functionals whose integrands depend not only on spatial position but also on the local tangent direction of the sensing path, which prevents a direct application of classical shape-calculus tools. To address this issue, we develop a shape-derivative framework that explicitly accounts for this tangent dependence, enabling gradient-based, non-parametric shape optimization of sensor geometries. In particular, we derive shape derivatives of T-and D-optimality criteria associated with the Fisher information matrix, as well as their counterparts based on the modified constitutive relation error. The resulting expressions extend classical Hadamard formulas and provide a rigorous basis for continuous, gradient-driven optimization of distributed sensor paths, allowing efficient and systematic sensor placement without relying on predefined geometric parameterizations.