Characterizing viscoelastic materials via ensemble-based data assimilation of bubble collapse observations

Abstract

Viscoelastic material properties at high strain rates are needed to model many biological and medical systems. Bubble cavitation can induce such strain rates, and the resulting bubble dynamics are sensitive to the material properties. Thus, in principle, these properties can be inferred via measurements of the bubble dynamics. Estrada et al. (2018) demonstrated such bubble-dynamic high-strain-rate rheometry by using least-squares shooting to minimize the difference between simulated and experimental bubble radius histories. We generalize their technique to account for additional uncertainties in the model, initial conditions, and material properties needed to uniquely simulate the bubble dynamics. Ensemble-based data assimilation minimizes the computational expense associated with the bubble cavitation model, providing a more efficient and scalable numerical framework for bubble-collapse rheometry. We test an ensemble Kalman filter (EnKF), an iterative ensemble Kalman smoother (IEnKS), and a hybrid ensemble-based 4D-Var method (En4D-Var) on synthetic data, assessing their estimations of the viscosity and shear modulus of a Kelvin-Voigt material. Results show that En4D-Var and IEnKS provide better moduli estimates than EnKF. Applying these methods to the experimental data of Estrada et al. (2018) yields similar material property estimates to those they obtained, but provides additional information about uncertainties. In particular, the En4D-Var yields lower viscosity estimates for some experiments, and the dynamic estimators reveal a potential mechanism that is unaccounted for in the model, whereby the apparent viscosity is reduced in some cases due to inelastic behavior, possibly in the form of material damage occurring at bubble collapse.

Keywords: A. dynamics; B. constitutive behaviour; B. viscoelastic material; C. numerical algorithms; data assimilation.

Figure 1:

Simulated bubble radius and noisy sampled data used to test data assimilation methods (a), alongside simulated bubble-wall velocity, normalized bubble pressure and stress integral (b), plotted over non-dimensional time t∗.

Figure 2:

Estimation of shear modulus and viscosity with initial guesses of G = 3.8 kPa and μ = 0.05Pas (both at 50% error). The estimation is plotted over non-dimensional time t for the EnKF and IEnKS methods, and over iteration number for the En4D–Var.

Figure 3:

Histogram of the final estimate for log(Ca) with the lag 1 IEnKS and fitted normal curve, where n is the number of ensemble members at each value of log(Ca).

Figure 4:

Comparing final ensembles for log(Ca) (a) and log(Re) (b) in the case with 50% initial error in both parameters.

Figure 5:

En4D–Var results for G (a) and μ (b), for ten simulated data sets.

Figure 6:

Histogram for G combining 10 final ensembles for simulated data runs with En4D-Var.

Figure 7:

Radius curve given by En4D–Var estimates and experimental measurements for data set 10.

Figure 8:

En4D–Var estimates for 10 experimental data sets.

Figure 9:

Comparing final combined ensembles for viscosity estimation in simulated and experimental data.

Figure 10:

Comparing online estimation of viscosity in data sets 2 (a) and 3 (b) using the IEnKS–MDA.

Figure 11:

Bar plot of radius normalized root mean squared errors for each data set. Also plotted are the previous estimate mean NRMSE (mean of all sets), the mean NRMSE for sets 3 to 5, and the final estimate mean NRMSE (mean of all other sets).