Time-domain inverse modeling of Euler–Bernoulli beams supported by an elastic foundation

Abstract

This paper presents a full-waveform inversion (FWI) method for reconstructing the material properties of Euler–Bernoulli beams supported by an elastic foundation, based on measured beam deflections. The forward problem is formulated as an initial–boundary-value problem, incorporating the Euler–Bernoulli beam equation and a Winkler foundation model in the time domain. A PDE-constrained optimization framework is employed to solve the inverse problem, aiming to minimize the misfit between measured and computed deflections, with regularization terms for control variables. The material profiles to be reconstructed include the elastic modulus of the beam and the stiffness of the Winkler foundation. The optimization process involves augmenting the objective functional into a Lagrangian functional and enforcing its stationarity to iteratively solve the state, adjoint, and control problems. Numerical examples examine three cases: (1) reconstruction of the beam’s elastic modulus with known foundation stiffness, (2) reconstruction of the foundation stiffness with known beam modulus, and (3) simultaneous reconstruction of both properties. Results show that both profiles can be accurately reconstructed simultaneously using a single set of measured deflections, without requiring subjective intervention. The developed FWI method demonstrates potential applications in nondestructive evaluation and health monitoring of lifeline structures, including underground pipelines and pavements, by enabling the accurate reconstruction of mechanical properties in beam–foundation systems.