Inverse modeling and experimental validation for reconstructing wave sources on a 2D solid from surficial measurement

Stephen Lloyd, Christoph Schaal, Chanseok Jeong

Research output: Contribution to journalArticlepeer-review


This paper discusses a source inversion method for the reconstruction of moving or stationary wave sources on the top surface of a two-dimensional (2D) linear elastic solid. This adjoint-gradient-based source inversion method uses vibrational measurements from sensors at the top surface of the solid, which can be heterogeneous and damped, to reconstruct temporal and spatial distributions of the wave sources. The finite element method (FEM) is used to obtain wave solutions with the high-resolution discretization of source functions in space and time leading the number of inversion parameters to range in the millions. Numerical experiments, in which the iterative inversion procedure begins with an initial guess of zero loading at all points in space and time, show that the presented approach is effective at reconstructing horizontal and vertical components of force (i.e., normal and shear tractions) for multiple simultaneous moving dynamic distributed loads without any prior knowledge about the loads except that all loading is applied along the top surface of the solid. Provided that moving loads on roadways are applied to the top surface, it is shown that updating the guessed loading at just surface nodes, rather than at all nodes in space, greatly improves the inversion results. The inversion is shown to be as effective at reconstructing loads on the top surface of a solid when the solid is horizontally layered with multiple materials as when the solid it is homogeneous. Reducing the distance between sensors improves the accuracy of the inversion while reducing the width of distributed loads leads to less accurate results. The authors also validate the presented inversion method by using experimental data obtained from lab-scale tests at a high frequency (100 kHz) for a stationary load on a homogeneous solid.

Original languageEnglish
Article number106880
StatePublished - Feb 2023


  • Discretize-then-optimize (DTO) approach
  • Experimental validation
  • Moving loads
  • Smart road/city
  • Wave source inversion
  • Weigh-in-motion


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