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FFT-based surrogate modeling of auxetic metamaterials with real-time prediction of effective elastic properties and swift inverse design

Hooman Danesh, Daniele Di Lorenzo, Francisco Chinesta, Stefanie Reese, Tim Brepols

TL;DR

This work addresses the computational bottleneck in predicting and designing auxetic metamaterials by combining FFT-based homogenization with data-driven surrogates. By generating a large high-fidelity dataset (≈$48{,}000$ samples on a $256\times256$ grid) for four orthogonal-void unit cells and training random-forest models to predict the normalized effective tangential stiffness components $\{C_{11}/E,\ C_{12}/E,\ C_{33}/E\}$, the authors enable real-time evaluation and rapid inverse design with $R^2$ values exceeding 0.999 on test data. They demonstrate parameter-space exploration, an inverse-design brute-force approach, and a GUI that facilitates both real-time prediction and inverse parameter retrieval, with FE validation confirming high accuracy (e.g., $\nu_{\mathrm{eff}}$ matching to three decimals and $C_{11}$/ $C_{12}$ within fractions of a percent). The methodology provides a practical tool for tailoring auxetic responses in engineering applications and suggests avenues for extending surrogate models to full-field outputs and active-learning data generation.

Abstract

Auxetic structures, known for their negative Poisson's ratio, exhibit effective elastic properties heavily influenced by their underlying structural geometry and base material properties. While periodic homogenization of auxetic unit cells can be used to investigate these properties, it is computationally expensive and limits design space exploration and inverse analysis. In this paper, surrogate models are developed for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids of different shapes. The unit cells feature orthogonal voids in four distinct shapes, including rectangular, diamond, oval, and peanut-shaped voids, each characterized by specific void diameters. The generated surrogate models accept geometric parameters and the elastic properties of the base material as inputs to predict the effective elastic constants in real-time. This rapid evaluation enables a practical inverse analysis framework for obtaining the optimal design parameters that yield the desired effective response. The fast Fourier transform (FFT)-based homogenization approach is adopted to efficiently generate data for developing the surrogate models, bypassing concerns about periodic mesh generation and boundary conditions typically associated with the finite element method (FEM). The performance of the generated surrogate models is rigorously examined through a train/test split methodology, a parametric study, and an inverse problem. Finally, a graphical user interface (GUI) is developed, offering real-time prediction of the effective tangent stiffness and performing inverse analysis to determine optimal geometric parameters.

FFT-based surrogate modeling of auxetic metamaterials with real-time prediction of effective elastic properties and swift inverse design

TL;DR

This work addresses the computational bottleneck in predicting and designing auxetic metamaterials by combining FFT-based homogenization with data-driven surrogates. By generating a large high-fidelity dataset (≈ samples on a grid) for four orthogonal-void unit cells and training random-forest models to predict the normalized effective tangential stiffness components , the authors enable real-time evaluation and rapid inverse design with values exceeding 0.999 on test data. They demonstrate parameter-space exploration, an inverse-design brute-force approach, and a GUI that facilitates both real-time prediction and inverse parameter retrieval, with FE validation confirming high accuracy (e.g., matching to three decimals and / within fractions of a percent). The methodology provides a practical tool for tailoring auxetic responses in engineering applications and suggests avenues for extending surrogate models to full-field outputs and active-learning data generation.

Abstract

Auxetic structures, known for their negative Poisson's ratio, exhibit effective elastic properties heavily influenced by their underlying structural geometry and base material properties. While periodic homogenization of auxetic unit cells can be used to investigate these properties, it is computationally expensive and limits design space exploration and inverse analysis. In this paper, surrogate models are developed for the real-time prediction of the effective elastic properties of auxetic unit cells with orthogonal voids of different shapes. The unit cells feature orthogonal voids in four distinct shapes, including rectangular, diamond, oval, and peanut-shaped voids, each characterized by specific void diameters. The generated surrogate models accept geometric parameters and the elastic properties of the base material as inputs to predict the effective elastic constants in real-time. This rapid evaluation enables a practical inverse analysis framework for obtaining the optimal design parameters that yield the desired effective response. The fast Fourier transform (FFT)-based homogenization approach is adopted to efficiently generate data for developing the surrogate models, bypassing concerns about periodic mesh generation and boundary conditions typically associated with the finite element method (FEM). The performance of the generated surrogate models is rigorously examined through a train/test split methodology, a parametric study, and an inverse problem. Finally, a graphical user interface (GUI) is developed, offering real-time prediction of the effective tangent stiffness and performing inverse analysis to determine optimal geometric parameters.
Paper Structure (17 sections, 31 equations, 14 figures, 2 tables)

This paper contains 17 sections, 31 equations, 14 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Context and rationale
  3. In silico data generation
  4. Objectives and contributions
  5. FFT-based homogenization
  6. Homogenized effective tangent stiffness
  7. Surrogate modeling
  8. Input parameters
  9. Output quantities
  10. Random forests regression
  11. Inverse analysis
  12. Case studies
  13. A parametric study
  14. Dataset size
  15. An inverse problem
  16. ...and 2 more sections

Figures (14)

  • Figure 1: Auxetic unit cells with orthogonal voids of different shapes, characterized by the unit cell length $L$ and void diameters $d$ and $D$.
  • Figure 2: Plots of the predicted values (obtained from the surrogate models) versus the actual values (obtained from the FFT-based solver) of the normalized effective elastic constants $C_{11}/E$, $C_{12}/E$, and $C_{33}/E$ for the rectangular void unit cell. The diagonal dashed line depicts the perfect prediction.
  • Figure 3: Plots of the predicted values (obtained from the surrogate models) versus the actual values (obtained from the FFT-based solver) of the normalized effective elastic constants $C_{11}/E$, $C_{12}/E$, and $C_{33}/E$ for the diamond void unit cell. The diagonal dashed line depicts the perfect prediction.
  • Figure 4: Plots of the predicted values (obtained from the surrogate models) versus the actual values (obtained from the FFT-based solver) of the normalized effective elastic constants $C_{11}/E$, $C_{12}/E$, and $C_{33}/E$ for the oval void unit cell. The diagonal dashed line depicts the perfect prediction.
  • Figure 5: Plots of the predicted values (obtained from the surrogate models) versus the actual values (obtained from the FFT-based solver) of the normalized effective elastic constants $C_{11}/E$, $C_{12}/E$, and $C_{33}/E$ for the peanut-shaped void unit cell. The diagonal dashed line depicts the perfect prediction.
  • ...and 9 more figures