Leveraging a data-driven approach to simulate and experimentally validate a MIMO multiphysics vibroacoustic system

Highlights

• To present a novel data-driven modeling technique using the vector fitting algorithm for simulating a multiphysics dynamic environment.

• To use the data driven model to determine the ideal forcing parameters for generating acoustic and structural traveling wave.

• To validate the data-driven model by experimentally generating acoustic and structural traveling wave.

Abstract

The objective of the present work is to evaluate the performance of a data-driven approach to simulate a multiphysics vibroacoustic field. Of particular interest is the complex propagation of anechoic waves in a finite 1D-structure submerged in water. Experimental data from an underwater beam actuated by two macro-fiber composites (MFCs) serves as the experimental framework for the data-driven model. Next, the Vector-Fitting (VF) algorithm estimates a state-space model of the multiphysics dynamics with voltage signals to the two MFCs as inputs, and the structural and the acoustic responses as the outputs. The multi-input–multi-output data-driven model is then used to determine the parameters that result in the optimal anechoic wave, a process that was carried out previously by experimental iterations. The optimal time-domain simulations are validated with experimental results.

Introduction

Multiphysics models analyze the interactions between multiple coupled simultaneous physical phenomena. Physics-based models are commonly used to numerically simulate such coupling. However, developing accurate physics-based models is not always feasible or straightforward. In such situations, are data-driven models a viable alternative? How robust are data-driven models in representing complex multiphysics interaction, as in the case of a structure vibrating underwater? The current effort addresses these questions and evaluates the performance of a data-driven modeling approach. The vector-fitting (VF) [1] algorithm is explored as an alternative to traditional numerical/computational means of simulating fluid–structure interaction.

Broadly, most models are used to simulate and predict the response $\left[x\left(t\right)\right]$ of a dynamical system to a given input excitation $\left[u\left(t\right)\right]$. In the Laplace domain, the response $\left[x\left(s\right)\right]$ of a linear, time-invariant dynamic system to the input $\left[u\left(s\right)\right]$ can be expressed in terms of the system’s characteristic transfer function $\left[\mathcal{H}\left(s\right)\right]$ as $\left[x\left(s\right)\right]=\left[\mathcal{H}\left(s\right)\right]\left[u\left(s\right)\right]$. While the physics-based modeling approach makes use of first-principles to arrive at a model that determines $\left[\mathcal{H}\left(s\right)\right]$, the data-driven approach estimates an approximate model from a experimentally sampled $\left[\mathcal{H}\left(i\omega \right)\right]$. In the vibroacoustic perspective, the transfer function $\left[\mathcal{H}\left(s\right)\right]$ of a $n$ degree-of-freedom linear system can be represented by its poles (${\lambda }_i$) - residues ($\left[{\mathit{R}}_i\right]$) as

$${\left[\mathcal{H}\left(s\right)\right]}_{(n\times n)}=\sum _{k=1}^n\left(\frac{{\left[{\mathit{R}}_k\right]}_{(n\times n)}}{s-{\lambda }_k}+\frac{{\left[{\mathit{R}}_k^{\ast }\right]}_{(n\times n)}}{s-{\lambda }_k^{\ast }}\right),$$

where ${\left\{\cdot \right\}}^{\ast }$ denotes the complex conjugate operation. The theoretical frequency response function (FRF) matrix ${\left[\mathcal{H}\left(i\omega \right)\right]}_{(n\times n)}$ is produced by sampling this continuous function at discrete frequencies $s=i\omega$. For a ‘$p$’ input - ‘$q$’ output experiment, a subset of the full FRF matrix is generated ${\left[\mathcal{H}\left(i\omega \right)\right]}_{(p\times q)}$ at each frequency, where $p\le n$ and $q\le n$. Consequently, the inverse problem aims to develop a data-driven model $\tilde{\mathcal{H}}\left(i\omega \right)$ by estimating poles ${\tilde{\lambda }}_k$ and residues ($\left[{\tilde{\mathit{R}}}_k\right]$). This results in a nonlinear least-squares problem that minimizes the error function $\varepsilon$ given by

$$\varepsilon ={\Vert {\left[\mathcal{H}\left(i\omega \right)\right]}_{(p\times q)}-\sum _{k=1}^n\left(\frac{{\left[{\tilde{\mathit{R}}}_k\right]}_{(p\times q)}}{i\omega -{\tilde{\lambda }}_k}+\frac{{\left[{\tilde{\mathit{R}}}_k^{\ast }\right]}_{(p\times q)}}{i\omega -\widetilde{{\lambda }_k^{\ast }}}\right)\Vert }_2.$$

This inverse problem can be challenging to solve for large-experimental data sets with significant numerical noise. Over the last few decades, numerous system identification techniques have adopted different approaches for solving the nonlinear least-squares problem by approximating the error function (2) to a linear least-squares problem. Each approach uses different forms of the transfer function shown in (1), to arrive at a linearized variant of (2). Interested readers are refereed to [2], [3] for details on various modal identification methods. One popular approach is to rewrite each FRF in (1) as a ratio of two polynomials i.e., $\tilde{\mathcal{H}}\left(i\omega \right)=\mathrm{N}\left(i\omega \right)/\mathrm{D}\left(i\omega \right)$ and as a result, the error function is linearized as

$${\varepsilon }^{\text{lin}}=\varepsilon {\Vert \mathrm{D}\left(i\omega \right)\Vert }_2={\Vert \mathcal{H}\left(i\omega \right)\mathrm{D}\left(i\omega \right)-\mathrm{N}\left(i\omega \right)\Vert }_2.$$

However, (3) is not an accurate simplification of (2), as ${\Vert \mathrm{D}\left(i\omega \right)\Vert }_2$ is not constant over the entire frequency spectrum. Such linear least-squares approximations are inaccurate representations of the inherent dynamical system. Furthermore, writing the transfer function as a polynomial could lead to ill-conditioned Vandermonde matrices. The VF algorithm overcomes these issues by relocating the starting poles and improves the pole-location based on the FRF data while preserving the nonlinear rational form of the FRF. The VF algorithm adopts a barycentric form of the transfer function  [1], which sets up an inverse problem using simple fractions resulting in a numerically stable algorithm. Other recent algorithms such as the adaptive Antoulas–Anderson algorithm (AAA) have performance comparable to the VF algorithm. Interested readers are referred to [4], [5], [6] for an extensive study comparing the VF and the AAA algorithms and [7], [8] for a discussion on the differences between the VF algorithm and other traditional vibroacoustic methods.

Additionally, in a previous effort [7], authors used a VF-based data-driven methodology to estimate the dispersion characteristics of a structural medium from the steady-state frequency responses. The phase and the group velocities of waves were determined over a broad frequency bandwidth of $50$kHz from transient simulations. The focus of that study was to determine the ensemble wave propagation characteristics rather than the time-domain response of individual waves. Consequently, the current work extends this methodology to a multiphysics vibroacoustic application, where data-driven models are developed for time-domain simulations.

As part of the current study, steady-state vibroacoustic experiments generate FRFs, and the VF algorithm uses these  FRFs to produce a state-space representation of the coupled dynamical system. The state-space model is then used to carry out a parametric study investigating the optimal parameters to generate structural and acoustic traveling waves on the structure and in the near surrounding fluid environment, respectively. The VF algorithm’s capability to accurately simulate the time-domain response of the multiphysics system is then validated against anechoic traveling wave experiments. Although there are numerous system identification algorithms in the vibroacoustic domain that study the inverse problem of estimating poles and residues of a dynamical system, there exists limited literature in evaluating the performance of the resulting data-driven models in simulating time-domain response. One of the focus areas of the current case study is to assess the capabilities of such data-driven models. Although the VF algorithm is the choice of system identification algorithm in the current study, other similar algorithms can be employed in developing similar data-driven models.

A one-dimensional beam submerged in water is used as an example to study the resulting underwater vibroacoustic field. A data-driven model is built to reproduce the frequency-domain response and the time-domain response to a multi-input excitation. Previously developed [9], [10] multi-input forcing technique induces steady-state traveling/progressing waves in the beam. These waves are known as anechoic waves, as they appear to “not reflect” (or disappear) at structural boundaries. The phase difference between the input forces is an important parameter for controlling the quality of the anechoic waves. Often, simulations based on finite element models or analytical models aid in determining the optimal phase. When these models are not readily available, as in the case of complex multi-physics systems, it necessitates rigorous experimental iterations. Thus, in this paper, a data-driven modeling approach is adopted to determine the optimal phase, yielding anechoic waves — structural waves and acoustic waves. The scope of the present study is limited to studying the acoustic traveling waves induced by the structural beam in the beam’s immediate vicinity. Although dynamics of the enclosure exist in the water basin, they do not show up in the frequency band selected.