Adjoint-based recovery of thermal fields from displacement or strain measurements

Abstract

A finite-element method dependent adjoint-based procedure to determine the temperature field of structures based on measured displacements/strains and a set of standard loads is developed and tested. Given a series of force and deformation measurements, the temperature field is obtained by minimizing the adequately weighted differences between the measured and computed values. Three numerical examples — a Plate With a Hole, a Bridge, and a Hoover Dam example — each with multiple sensors distributed in different configurations, demonstrate the procedure’s capabilities. A target temperature distribution is prescribed in all cases, and the displacement sensor data is recorded. The optimization algorithm (here, steepest descent with Barzilai–Borwein step) uses this data to optimize the temperatures such that the same deformation is obtained at the sensor locations. Vertex Morphing is used as a filter to mitigate the ill-conditioning. Results show that the proposed approach can accurately reconstruct the target thermal distribution, especially when more sensors are used. Additionally, it is observed that the sensors do not need to be positioned in the region of interest; the method remains effective as long as the sensors can detect changes related to that area. A comparison with standard spatial interpolation techniques, namely, k-nearest neighbors and ordinary and universal kriging, is performed using temperature sensors in the same configurations. The proposed approach performs remarkably better than the interpolation techniques with a reduction in the $L2$ norm of up to 41.3%, 93.9%, and 41.3%, for the Plate With a Hole, the Bridge, and the Dam examples, respectively.