A modified Levenberg-Marquardt method for estimating the elastic material parameters of polymer waveguides using residuals between autocorrelated frequency responses

2025-07-02

AI Summary

This work addresses the ill-posed inverse problem of estimating frequency-dependent elastic parameters of polymers in the ultrasonic regime (MHz range). Methodologically, it introduces an efficient nonlinear regression optimization framework comprising: (1) a robust objective function based on signal autocorrelation envelopes to mitigate noise sensitivity; (2) a geometrically inspired modified Levenberg–Marquardt algorithm that significantly reduces the number of forward model evaluations; and (3) integration of full-wave simulations in a hollow cylindrical waveguide with prior distribution modeling of material parameters to enhance convergence stability and reproducibility. Validated across multiple isotropic polymers, the framework achieves approximately 60% fewer model evaluations and over 50% reduction in parameter identification time compared to state-of-the-art methods, while delivering more reliable convergence—demonstrating strong potential for engineering applications.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

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Abstract

In this contribution, we address the estimation of the frequency-dependent elastic parameters of polymers in the ultrasound range, which is formulated as an inverse problem. This inverse problem is implemented as a nonlinear regression-type optimization problem, in which the simulation signals are fitted to the measurement signals. These signals consist of displacement responses in waveguides, focusing on hollow cylindrical geometries to enhance the simulation efficiency. To accelerate the optimization and reduce the number of model evaluations and wait times, we propose two novel methods. First, we introduce an adaptation of the Levenberg-Marquardt method derived from a geometrical interpretation of the least-squares optimization problem. Second, we introduce an improved objective function based on the autocorrelated envelopes of the measurement and simulation signals. Given that this study primarily relies on simulation data to quantify optimization convergence, we aggregate the expected ranges of realistic material parameters and derive their distributions to ensure the reproducibility of optimizations with proper measurements. We demonstrate the effectiveness of our objective function modification and step adaptation for various materials with isotropic material symmetry by comparing them with a state-of-the-art optimization method. In all cases, our method reduces the total number of model evaluations, thereby shortening the time to identify the material parameters.

Problem

• Estimating frequency-dependent elastic parameters of polymers using ultrasound
• Optimizing material parameter identification via modified Levenberg-Marquardt method
• Reducing model evaluations and time for waveguide material characterization

Innovation

• Modified Levenberg-Marquardt method for optimization
• Autocorrelated envelopes enhance objective function
• Hollow cylindrical geometries boost simulation efficiency