Physics-Constrained Data-Driven Variational Method for Discrepancy Modeling

This paper presents a data-driven discrepancy modeling method that variationally embeds measured data in the modeling and analysis framework. The proposed method exploits the residual between the first-principles theory and sensor-based measurements from the dynamical system, and it augments the physics-based model with a variationally derived loss function that is comprised of this residual. The method was first developed in the context of linear elasticity (Masud and Goraya, J. Appl. Mech. 89 (11), 111001 (2022)) wherein the relation between the discrepancy model and loss terms was derived to show that the data embedding terms behave like residual-based least-squares regression functions. An interpretation of the stabilization tensor as a kernel function was formally established and its role in assimilating a-priori knowledge of the problem in the modeling method was highlighted. The present paper employs linear elastodynamics as a model problem where the Data-Driven Variational (DDV) method incorporates high-fidelity data into the forward simulations, thereby driving the problem with not only the boundary and initial conditions, but also by measurement data that is taken at only a small subset of the total domain. The effect of the loss function on the time-dependent response of the system is investigated under a variety of loading conditions and model discrepancies. The energy and Morlet wavelet analyses reveal that the problem with embedded data recovers the energy and the fundamental frequency band of the target system. Time histories of strain energy and kinetic energy of a cantilever beam undergoing damped oscillations are recovered by including known data in an undamped model to highlight the data-driven discrepancy modeling feature of the method under the combined effect of parameter and model discrepancy.

Inversion algorithm for Lamb-wave-based depth characterization of acoustic emission sources in plate-like structures

An inversion algorithm (termed AEDep) is proposed for estimating the depth of acoustic emission (AE) sources in plate-like structural components. The work is motivated by the need for characterizing early-stage fatigue crack growth in such components. The algorithm achieves depth estimation by automatically extracting the depth-dependent amplitude ratio between the fundamental Lamb modes which comprise the AE signals. A finite element model is designed to study the frequency-dependent forward problem of Lamb wave motion due to a given source, from which the relation between source depth and amplitude ratio is established. Elastodynamic theory is used to validate the model in the frequency domain, as well as to derive a sensor tuning factor which may be incorporated into the solution. The proposed algorithm was tested on two plate-like specimens: a 6061-T6 aluminum plate and a 2025-T6 aluminum aircraft fuselage panel. Validation of the algorithm was achieved by generating controlled AE sources at various depths along the edges of the specimens, in the form of Hsu-Nielsen pencil lead breaks. Good agreement was found in the aluminum plate between the true and estimated source depths. A slight decrease in accuracy was found in the fuselage panel between the true values and their estimations. However, both experimental cases demonstrated the ability to distinguish between sources originating near the mid-plane of a plate-like structure from those near the surface. Lastly, the fast computation of the inversion algorithm shows strong potential for real-time monitoring applications.

Topology preserving non-rigid image registration using time-varying elasticity model for MRI brain volumes

In this paper, we present a new non-rigid image registration method that imposes a topology preservation constraint on the deformation. We propose to incorporate the time varying elasticity model into the deformable image matching procedure and constrain the Jacobian determinant of the transformation over the entire image domain. The motion of elastic bodies is governed by a hyperbolic partial differential equation, generally termed as elastodynamics wave equation, which we propose to use as a deformation model. We carried out clinical image registration experiments on 3D magnetic resonance brain scans from IBSR database. The results of the proposed registration approach in terms of Kappa index and relative overlap computed over the subcortical structures were compared against the existing topology preserving non-rigid image registration methods and non topology preserving variant of our proposed registration scheme. The Jacobian determinant maps obtained with our proposed registration method were qualitatively and quantitatively analyzed. The results demonstrated that the proposed scheme provides good registration accuracy with smooth transformations, thereby guaranteeing the preservation of topology.

Depth profile of a time-reversal focus in an elastic solid

The out-of-plane velocity component is focused on the flat surface of an isotropic solid sample using the principle of time reversal. This experiment is often reproduced in the context of nondestructive testing for imaging features near the surface of the sample. However, it is not clear how deep the focus extends into the bulk of the sample and what its profile is. In this paper, this question is answered using both numerical simulations and experimental data. The profiles of the foci are expressed in terms of the wavelengths of the dominant waves, based on the interpretation of the Lamb's problem and the use of the diffraction limit.