Fisher-Information-Based Sensor Placement for Structural Digital Twins: Analytic Results and Benchmarks
Harbir Antil, Animesh Jain, Rainald Löhner
TL;DR
This work develops a rigorous Fisher-information-based framework for sensor placement in adjoint-based finite-element structural digital twins. It provides matrix-free operator formulas to compute Fisher-information products F = J^T R^{-1} J and derives explicit sensitivities of the D-optimal objective with respect to sensor design, relying only on forward/adjoint solves and measurement operators. Analytically tractable 1D benchmarks are offered to distinguish detectability from localizability and to prove that D-optimal placement of multiple displacement sensors yields approximately uniform spacing, including closed-form designs for several sensor counts. The methodology enables offline, information-driven sensor configuration within FE-based inverse solvers, with practical steps for gradient computation and robust design options. The results bridge theory and practice, informing sensor deployment strategies that maximize identifiability and localization in structural health monitoring and digital twins.
Abstract
High-fidelity digital twins rely on the accurate assimilation of sensor data into physics-based computational models. In structural applications, such twins aim to identify spatially distributed quantities--such as elementwise weakening fields, material parameters, or effective thermal loads--by minimizing discrepancies between measured and simulated responses subject to the governing equations of structural mechanics. While adjoint-based methods enable efficient gradient computation for these inverse problems, the quality and stability of the resulting estimates depend critically on the choice of sensor locations, measurement types, and directions. This paper develops a rigorous and implementation-ready framework for Fisher-information-based sensor placement in adjoint-based finite-element digital twins. Sensor configurations are evaluated using a D-optimal design criterion derived from a linearization of the measurement map, yielding a statistically meaningful measure of information content. We present matrix-free operator formulas for applying the Jacobian and its adjoint, and hence for computing Fisher-information products $Fv = J^\top R^{-1} Jv$ using only forward and adjoint solves. Building on these operator evaluations, we derive explicit sensitivity expressions for D-optimal sensor design with respect to measurement parameters and discuss practical strategies for evaluating the associated log-determinant objectives. To complement the general framework, we provide analytically tractable sensor placement results for a canonical one-dimensional structural model, clarifying the distinction between detectability and localizability and proving that D-optimal placement of multiple displacement sensors yields approximately uniform spacing.