Ensemble WSINDy for Data Driven Discovery of Governing Equations from Laser-based Full-field Measurements

Abstract

This work leverages laser vibrometry and the weak form of the sparse identification of nonlinear dynamics (WSINDy) for partial differential equations to learn macroscale governing equations from full-field experimental data. In the experiments, two beam-like specimens, one aluminum and one IDOX/Estane composite, are subjected to shear wave excitation in the low frequency regime and the response is measured in the form of particle velocity on the specimen surface. The WSINDy for PDEs algorithm is applied to the resulting spatio-temporal data to discover the effective dynamics of the specimens from a family of potential PDEs. The discovered PDE is of the recognizable Euler-Bernoulli beam model form, from which the Young's modulus for the two materials are estimated. An ensemble version of the WSINDy algorithm is also used which results in information about the uncertainty in the PDE coefficients and Young's moduli. The discovered PDEs are also simulated with a finite element code to compare against the experimental data with reasonable accuracy. Using full-field experimental data and WSINDy together is a powerful non-destructive approach for learning unknown governing equations and gaining insights about mechanical systems in the dynamic regime.

Figures (10)

• Figure 1: Experimental set-up and sensing configuration diagram. The IE sample is vertically mounted on a shear-wave transducer inducing the five-cycle burst input. The resulting wave motion is then measured in the form of transverse particle velocity by a laser Doppler vibrometer scanning on a uniform grid of $113$ points, with $0.5$ increments, along the beam length.
• Figure 2: Experimental data. Each panel shows the velocity measurements (colorbar) in time (y-axis) and space (x-axis). The aluminum data is in the top row and the IE is in the bottom row. The filtered experimental data is shown in (a) and the truncated data used for equation discovery is shown in (b).
• Figure 3: Natural frequency vs mode for Al and IE beams computed using the testing configuration, the geometry of specimens and $E$ values in Table \ref{['tab:eqdiscovery']}. The Al results are shown in blue and the IE results in red. The input frequencies are marked with the dashed lines and fall within the first one to two modes for each specimen.
• Figure 4: Al FEM results for two values of $E$. The experimental field is shown in (a), field simulated with FEM is shown in (b), and a error field computed via $|\mathbf{W}-\hat{\mathbf{W}}|$ is shown in (c). The top row corresponds to $E=6.3206e+10$ Pa found from the WSINDy coefficient and the bottom row corresponds to $E=6.33056e+10$ Pa found from minimizing the relative Frobenius error.
• Figure 5: IE FEM results for two values of $E$. The experimental field is shown in (a), field simulated with FEM is shown in (b), and a error field computed via $|\mathbf{W}-\hat{\mathbf{W}}|$ is shown in (c). The top row corresponds to $E=9.6292e+8$ found from the WSINDy coefficient and the bottom row corresponds to $E=1.05912e+9$ Pa found from minimizing the relative Frobenius error.