PDE-constrained optimization for virtual sensing in structural dynamics: Full-field displacement and force recovery from sparse sensors

Virtual sensing -- recovering full-field structural response from sparse sensor measurements -- is a fundamental challenge in structural health monitoring. This study formulates virtual sensing as a PDE-constrained optimization (PDE-CO) problem, where the governing elastodynamic equation serves as an equality constraint, the applied force distribution is the optimization variable, and the full-field displacement and force are jointly recovered. Gradients of the Tikhonov-regularized objective are computed via reverse-mode automatic differentiation through the forward PDE solver, and L-BFGS iteratively finds the optimal state. The framework couples an offline FEniCSx stage for finite element assembly with an online GPU-accelerated JAX stage, and is verified on three examples of increasing complexity: a cantilever plate, a $90^\circ$ elbow pipe, and a reactor pressure vessel (RPV) representative of a 300~MW pressurized nuclear reactor. PDE-CO consistently outperforms modal expansion across all three cases, reducing displacement errors by factors of $2$ to $17\times$ with sub-percent accuracy on every example. Unlike modal expansion, where force is derived by back-calculation from truncated modal coordinates and is not jointly optimized, PDE-CO recovers displacement and force simultaneously through the PDE constraint; the increased computational cost is offset by GPU acceleration delivering up to $64.8\times$ speedup over CPU.