# Wave-based reduction of finite phononic structures with nonlinear boundary conditions

Vincent Mahé, Adrien Mélot, Benjamin Chouvion, Christophe Droz

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Vincent Mahé, Adrien Mélot, Benjamin Chouvion, Christophe Droz. Wave-based reduction of finite phononic structures with nonlinear boundary conditions. NODYCON 2025 - 4th International Nonlinear Dynamics Conference, Jun 2025, Hoboken, NJ, United States. pp.1-1. hal-05243653

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Submitted on 7 Sep 2025

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# Wave-based reduction of finite phononic structures with nonlinear boundary conditions

Vincent Mahé*, Adrien Mélot*, Benjamin Chouvion** and Christophe Droz*

*Inria

**Ecole de l'air et de l'espace

**Abstract.** This work introduces a novel computational methodology to compute the dynamic response of finite phononic structures. The methodology leverages wave-based modeling in order to reduce the problem to a single unit cell. The proposed methodology is very general and can be utilized to analyze finite element models of arbitrarily complex geometries. Its computational performance is validated with comparisons with conventional finite element models.

## Introduction

Structures constituted of assembled sub-components are often subject to nonlinear effects due to the presence of joints or contacts, which can induce higher-order harmonics of the source excitation. When the structure is periodic and nonlinearities are omitted, a common approach is to take advantage of Floquet-Bloch theory in order to reduce the whole model to a single unit cell. However, modern structures are increasingly operating in nonlinear regimes in order to improve performances [1][2] and classical Floquet-Bloch formulations cannot account for high order harmonics induced by nonlinearities. Existing methodologies to compute such nonlinear responses are only available for simple geometries with analytical solutions such as bars and beams. Thus, efficient mathematical modeling frameworks are needed.

## Results and Discussion

We propose here a multi-harmonic adaptation of the Wave Finite Element Method (WFEM) [3][4] in order to analyze finite phononic structures with nonlinear boundary conditions. The Harmonic Balance Method (HBM) [5] is employed in order to formulate the multi-harmonic wave problem as a system of nonlinear algebraic equations whose unknowns are the Bloch waves' amplitude resulting from the WFEM [6]. The nonlinear forces are evaluated via an alternating frequency-time method (AFT) and the frequency response of the system is computed using a continuation procedure. We illustrate the approach on the finite-length phononic structure shown in Fig 1, which features local resonators located within each unit cell and a nonlinear boundary condition. The proposed methodology results in a significant computational speedup. Moreover, we show that under certain conditions nonlinearities may shift a nonlinear resonance peak from outside to inside the locally-resonant bandgap and thus give rise to high amplitude solutions analogous to edge modes.

<figure><img src="image_2.png" /><figcaption>Figure 1: Schematic representation of the studied structure</figcaption></figure>

## References

[1] Mahé V., Renault A., Grolet A., Mahe H., Thomas O., (2022) Subharmonic centrifugal pendulum vibration absorbers allowing a rotational mobility. Mech. Syst. Signal Process., **177**:109125.

[2] Mélot A., Denimal E., Renson L. (2024) Multi-parametric optimization for controlling bifurcation structures. Proc. R. Soc. A: Math. Phys. Eng. Sci., **480**:2283.

[3] Mead D. (1973) A general theory of harmonic wave propagation in linear periodic systems with multiple coupling. J. Sound Vib., **27**:235-260.

[4] Droz C., Boukadia R., Desmet W. (2021) A multi-scale model order reduction scheme for transient modelling of periodic structures. J. Sound Vib., **510**:116312.

[5] Krack M., Gross J. (2019) Harmonic Balance for Nonlinear Vibration Problems. Springer Cham.

[6] Mahé V., Mélot A., Chouvion B., Droz C. (2025) Computing the dynamic response of periodic waveguides with nonlinear boundaries using the Wave Finite Element Method. Preprint.