Matching frequency response measurements and reduced order models for the inverse identification of viscoelastic properties

TL;DR

The paper addresses the challenge of identifying frequency-dependent viscoelastic properties for 3D-printed materials by integrating model-order reduction (MOR) with a four-parameter fractional derivative model. A second-order Taylor-based MOR around multiple expansion points, built on Krylov subspaces and TOAR, produces fast, global FRF surrogates; Sobol-driven DOE samples the parameter space and particle swarm optimization (PSO) estimates $E_0$, $E_{ extinfty}$, $\tau$, and $\alpha$ from FRF data. Validation against DMA for POM beams and application to curved 3D-printed polymer–ceramic patches demonstrate accurate FRF reproduction and robust parameter estimation, with peak frequencies within a few percent and damping estimates in the 5–7% range. The method enables rapid, data-driven viscoelastic identification from a single FRF measurement, offering practical advantages for complex geometries where DMA is impractical.

Abstract

3D-printed materials are used in many different industries (automotive, aviation, medicine, etc.). Most of these 3D-printed materials are based on ceramics or polymers whose mechanical properties vary with frequency. For numerical modeling, it is crucial to characterize this frequency dependency accurately to enable realistic finite-element simulations. At the same time, the damping behavior plays a key role in product development, since it governs a component's response at resonance and thus impacts both performance and longevity. In current research, inverse material characterization methods are getting more and more popular. However, their practical validation and applicability on real measurement data have not yet been discussed widely. In this work, we show the identification of two different materials, POM and additively manufactured sintered ceramics, and validate it with experimental data of a well-established measurement technique (dynamic mechanical analysis). The material identification process considers state-of-the-art reduced-order modeling and constrained particle swarm optimization, which are used to fit the frequency response functions of point measurements obtained by a laser Doppler vibrometer. This work shows the quality of the method in identifying the parameters defining the viscoelastic fractional derivative model, including their uncertainty. It also illustrates the applicability of this identification method in the presence of practical difficulties that come along with experimental data such as boundary conditions and noise.

Figures (14)

• Figure 1: Exemplary material behavior of the fractional derivative model, illustrating the storage modulus $E_s$ and loss modulus $E_l$.
• Figure 2: Left: DOE (normalized) illustration for quasi-random Sobol sampling points (blue) and validation points (red). Using the global basis, each validation point is compared to the full order model. Right: The material properties of the corresponding material table of the validation points.
• Figure 3: FRFs of the four validation points comparing the global MOR basis with the exact viscoelastic solution (left), and the relative error between the two solutions (right).
• Figure 4: Response surface visualization of the objective function (eq \ref{['eq:objectivefunc']}) for parameters $\alpha$ vs $\tau$ (a) and $\tau$ vs $E_0$ (b) of the polyoxymethylene beam shown in Sec. \ref{['sec:POM']} using 1 000 000 Latin hypercube sampling points. Several local minima with similar objective function values can be distinguished.
• Figure 5: Measured voltage to force response function (black dashed lines) and theoretical (normalized) approximation by a low pass filter with a cut-off frequency of 1040 Hz (orange line).