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Inverse Design of Periodic and Quasi-Periodic Nonlinear Mechanical Metamaterial — NODYCON 2023 Conference Paper Pravinkumar Ghodake (3701242) Engineering education Mechanical engineering asset management Mechanical engineering not elsewhere classified Acoustics and noise control (excl. architectural acoustics) Acoustics and acoustical devices; waves Solid mechanics Theoretical and applied mechanics Applications in physical sciences Other physical sciences not elsewhere classified Computational complexity and computability Computational modelling and simulation in earth sciences Numerical and computational mathematics not elsewhere classified Data visualisation and computational (incl. parametric and generative) design Numerical modelling and mechanical characterisation Nonlinear optics and spectroscopy Numerical analysis Numerical computation and mathematical software Numerical solution of differential and integral equations Engineering design Computer aided design Design practice and methods Industrial and product design Design not elsewhere classified Design history, theory and criticism Models and simulations of design Theory and design of materials NODYCON 2023 inverse design inverse design validity Inverse design model inverse design model inverse design approach inverse design process inverse design. inverse design approaches inverse design workflow inverse designed using inverse design problem inverse design optimization Inverse Design SimulationsAn Inverse Design ApproachDiscovery inverse design procedure inverse design bypasses periodic metamaterials quasi-periodic metamaterials nonlinear mechanical metamaterials nonlinear ultrasonics Nonlinear ultrasonics Nonlinear Ultrasonics harmonic generation harmonic generation light harmonic scattering Harmonic Scattering harmonic scattering intensities Harmonic Scattering Angle-resolved higher harmonics generation Higher harmonics higher harmonics second harmonic suppression third harmonic suppression Nelder-Mead algorithm Nelder-Mead Algorithm shape optimization shape optimization algorithms SHAPE OPTIMIZATION shape optimization computations Shape optimization Shape Optimization time-dependent optimization finite element method Finite element method -- Data processing finite element method model Finite element method - Data processing finite element method simulation Finite element method. layered metamaterials steel-glass metamaterials phononic crystals Phononic Crystals Phononic crystals bandgap system-generated harmonics power amplifier harmonics wave trapping Wave trapping behavior early-stage damage Early-Stage Damage Detection structural health monitoring Structural health monitoring systems Structural Health Monitoring Sensors Structural Health Monitoring (SHM). Structural health monitoring. Structural Health Monitoring (SHM) structural health monitoring (SHM) Structural health monitoring (SHM) Structural Health Monitoring STRUCTURAL HEALTH MONITORING Structural health monitoring structural health monitoring, data-driven methods, thermal response, bridges Structural health monitoring; Interfacial debonding; ACTs; Vibration-based Structural health monitoring Aircraft availability Structural health monitoring; Hybrid ceramic bearings; Systematic review; Meta-Analysis non-destructive evaluation non-destructive evaluation inspection Non-destructive evaluation (NDE) Non-Destructive Evaluation (NDE) Non-destructive Evaluation Non-Destructive Evaluation Non-destructive evaluation solid mechanics solid mechanics problems Solid mechanics and dynamics Solid mechanics Solid Mechanics Pravinkumar Ghodake Pravinkumar Ramchandra Ghodake Design Optimization Design & Technology Generative design research Elastodynamics simulations This conference contribution is a research paper presented by Pravinkumar Ghodake (Department of Mechanical Engineering, IIT Bombay) at NODYCON 2023 — the Third International Nonlinear Dynamics Conference. The work was submitted and published as a conference proceeding under the NODYCON Open Repository. This paper is paired with the companion NODYCON 2023 paper "Harmonic Scattering of Waves from Crossed-Thin-Rectangular Nonlinear Inclusions." Title of contribution: "Inverse Design of Periodic and Quasi-Periodic Nonlinear Mechanical Metamaterial" Abstract: In nonlinear ultrasonics, the interaction of a monochromatic wave with nonlinear elastic materials generates higher harmonics (2f, 3f, …), known as harmonic generation. The sensitivity of higher harmonic amplitudes due to harmonic generation and harmonic scattering towards design parameters of nonlinear metamaterials — such as widths of the nonlinear elastic layers — makes the inverse design of nonlinear metamaterials challenging. Periodic and quasi-periodic linear and nonlinear mechanical metamaterials are designed by solving inverse design problems to control nonlinear elastic waves in solids. The time-dependent inverse design problems include a forward problem implemented using the finite element method. The inverse problem is proposed as a time-dependent optimization problem and solved using the Nelder-Mead algorithm. Introduction: Nonlinear ultrasonics has shown effectiveness in quantifying early-stage damages such as micro-cracks, dislocation substructures, and micro-voids. Harmonic generation due to the interaction of monochromatic (f) waves with early-stage damages has been demonstrated in various theoretical, computational, and experimental studies. Measuring amplitudes of such higher harmonics is important, as they give information about the intensity of early-stage damages. During experiments, system-generated higher harmonics are introduced due to instrumentation, such as a short pulse power amplifier that masks the damaged material's natural harmonic response. Suppression of such system-generated higher harmonics, like 2nd (2f) and 3rd (3f) harmonics, can be achieved by designing appropriate metamaterials.Harmonic generation and scattering are highly sensitive to the widths of metallic layers present in metamaterials. The nonlinearity effects of the materials used in the design of metamaterials need to be addressed. This research aims to design robust linear and nonlinear metamaterials for nonlinear ultrasonic applications through the inverse design approach. A shape optimization problem is defined for the design of layered linear, nonlinear, periodic, and quasi-periodic metamaterials by simulating wave propagation studies using the finite element method. Results and Discussion:<ul><li>Capabilities of the inversely designed periodic and quasi-periodic linear and nonlinear metamaterials to control nonlinear waves are demonstrated through a novel, realistic, and universal inverse design approach.</li><li>Interestingly, some inversely designed linear metamaterials also show similar responses to nonlinear metamaterials.</li><li>Harmonically scattered waves are trapped within the metamaterial due to the complex interplay between harmonic generation and scattering of waves between multiple nonlinear elastic layers.</li><li>The inversely designed metamaterials suppress 2nd (2f) and 3rd (3f) harmonics while maintaining the amplitude of the 1st (f) harmonic at maximum.</li></ul> Key Contributions:<ul><li>Universal inverse design strategy for periodic and quasi-periodic nonlinear mechanical metamaterials</li><li>Time-dependent optimization formulation solved via the Nelder-Mead algorithm</li><li>Demonstration of harmonic suppression (2f, 3f) while preserving the fundamental (f)</li><li>Insight into wave trapping arising from harmonic generation and inter-layer scattering</li><li>Equivalence observed between some inversely designed linear and nonlinear metamaterials</li></ul> Methodology: A finite element method (FEM) framework is used to simulate wave propagation in layered periodic and quasi-periodic metamaterials composed of steel and glass layers. The forward problem is implemented in a time-dependent FEM setting, and the inverse problem is formulated as a time-dependent optimization problem solved using the gradient-free Nelder-Mead algorithm. Design variables include individual layer widths and the number of repeated unit cells. Perfectly matched layers (PMLs) and sensors S₁ and S₂ define the wave-input and wave-receiving boundaries. Conclusions: The proposed framework provides a universal approach to design linear and nonlinear mechanical metamaterials that control nonlinear elastic waves in solids. The results support advanced nonlinear ultrasonic testing applications by isolating true damage-induced harmonics from parasitic system-generated harmonics. Published abstract (NODYCON 2023 official): <a href="https://nodycon.org/2023/papers/190/abstract_submissions/569/view_abstract" target="_blank" rel="noreferrer">https://nodycon.org/2023/papers/190/abstract_submissions/569/view_abstract</a> Companion paper (NODYCON 2023): Harmonic Scattering of Waves from Crossed-Thin-Rectangular Nonlinear Inclusions — <a href="https://nodycon.org/2023/papers/337/abstract_submissions/581/view_abstract" target="_blank" rel="noreferrer">https://nodycon.org/2023/papers/337/abstract_submissions/581/view_abstract</a> Research portfolio: <a href="https://sites.google.com/view/pravinkumarghodake/research" target="_blank" rel="noreferrer">https://sites.google.com/view/pravinkumarghodake/research</a> References: [1] Kube C. M. (2017) Scattering of Harmonic Waves from a Nonlinear Elastic Inclusion. J. Acoust. Soc. Am., 141(6): 4756–4767. [2] Ghodake P. (2021) One Dimensional Nonlinear Wave Propagation in a Rate Independent Pinched Hysteretic Material. Proc. 48th Annu. Rev. Prog. in QNDE, 1–6. [3] Cantrell J. H., Yost W. (2001) Nonlinear Ultrasonic Characterization of Fatigue Microstructures. Int. J. Fatigue, 23: 487–490. [4] Liu S., Croxford A. J., Neild S. A., Zhou Z. (2011) Effects of Experimental Variables on the Nonlinear Harmonic Generation Technique. IEEE Trans Ultrason Ferroelectr Freq Control, 58(7): 1442–1451. Keywords: inverse design, periodic metamaterials, quasi-periodic metamaterials, nonlinear mechanical metamaterials, harmonic scattering, Nelder-Mead algorithm, shape optimization, finite element method, layered metamaterials, higher harmonics, NODYCON 2023, structural health monitoring 2026-09-30T05:16:27Z Text Conference contribution 10.6084/m9.figshare.34029912.v1 https://figshare.com/articles/conference_contribution/Inverse_Design_of_Periodic_and_Quasi-Periodic_Nonlinear_Mechanical_Metamaterial_NODYCON_2023_Conference_Paper/34029912 CC BY 4.0