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On the data-driven description of lattice materials mechanics

Ismael Ben-Yelun, Luis Irastorza-Valera, Luis Saucedo-Mora, Francisco Javier Montáns, Francisco Chinesta

TL;DR

The paper tackles the problem of rapidly predicting the effective stiffness $E_z^{eq}$ of lattice metamaterials across both periodic and aperiodic topologies to enable inverse design and multi-scale optimization. It introduces offline data generation of random 3D lattices via Delaunay triangulation and uses energy-based homogenization as ground truth, comparing a physics-informed dense neural network with a graph neural network that exploits the lattice connectivity. Results show that the GNN, with orders of magnitude fewer parameters, achieves near-perfect or very high $R^2$ values across pin-jointed and Euler-Bernoulli beam lattices, outperforming the DNN, and enabling fast forward evaluations for inverse design via Pareto optimization over volume and stiffness. The framework supports fast exploration of the design space, facilitating multi-objective optimization and potentially extending to non-linear or multi-material lattices, thereby significantly accelerating structure-property mappings in architectural metamaterials.

Abstract

In the emerging field of mechanical metamaterials, using periodic lattice structures as a primary ingredient is relatively frequent. However, the choice of aperiodic lattices in these structures presents unique advantages regarding failure, e.g., buckling or fracture, because avoiding repeated patterns prevents global failures, with local failures occurring in turn that can beneficially delay structural collapse. Therefore, it is expedient to develop models for computing efficiently the effective mechanical properties in lattices from different general features while addressing the challenge of presenting topologies (or graphs) of different sizes. In this paper, we develop a deep learning model to predict energetically-equivalent mechanical properties of linear elastic lattices effectively. Considering the lattice as a graph and defining material and geometrical features on such, we show that Graph Neural Networks provide more accurate predictions than a dense, fully connected strategy, thanks to the geometrically induced bias through graph representation, closer to the underlying equilibrium laws from mechanics solved in the direct problem. Leveraging the efficient forward-evaluation of a vast number of lattices using this surrogate enables the inverse problem, i.e., to obtain a structure having prescribed specific behavior, which is ultimately suitable for multiscale structural optimization problems.

On the data-driven description of lattice materials mechanics

TL;DR

The paper tackles the problem of rapidly predicting the effective stiffness of lattice metamaterials across both periodic and aperiodic topologies to enable inverse design and multi-scale optimization. It introduces offline data generation of random 3D lattices via Delaunay triangulation and uses energy-based homogenization as ground truth, comparing a physics-informed dense neural network with a graph neural network that exploits the lattice connectivity. Results show that the GNN, with orders of magnitude fewer parameters, achieves near-perfect or very high values across pin-jointed and Euler-Bernoulli beam lattices, outperforming the DNN, and enabling fast forward evaluations for inverse design via Pareto optimization over volume and stiffness. The framework supports fast exploration of the design space, facilitating multi-objective optimization and potentially extending to non-linear or multi-material lattices, thereby significantly accelerating structure-property mappings in architectural metamaterials.

Abstract

In the emerging field of mechanical metamaterials, using periodic lattice structures as a primary ingredient is relatively frequent. However, the choice of aperiodic lattices in these structures presents unique advantages regarding failure, e.g., buckling or fracture, because avoiding repeated patterns prevents global failures, with local failures occurring in turn that can beneficially delay structural collapse. Therefore, it is expedient to develop models for computing efficiently the effective mechanical properties in lattices from different general features while addressing the challenge of presenting topologies (or graphs) of different sizes. In this paper, we develop a deep learning model to predict energetically-equivalent mechanical properties of linear elastic lattices effectively. Considering the lattice as a graph and defining material and geometrical features on such, we show that Graph Neural Networks provide more accurate predictions than a dense, fully connected strategy, thanks to the geometrically induced bias through graph representation, closer to the underlying equilibrium laws from mechanics solved in the direct problem. Leveraging the efficient forward-evaluation of a vast number of lattices using this surrogate enables the inverse problem, i.e., to obtain a structure having prescribed specific behavior, which is ultimately suitable for multiscale structural optimization problems.
Paper Structure (11 sections, 13 equations, 2 figures)

This paper contains 11 sections, 13 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related research
  4. Materials and methods
  5. Lattice structures database
  6. Structure definition
  7. Boundary conditions and mechanical model of the lattice
  8. Boundary conditions and mechanical model of the lattice
  9. Effective behavior
  10. Dense Neural Networks model
  11. Prediction of springs arranged in series: a toy example

Figures (2)

  • Figure 1: Boundary conditions of the lattice structure, where $u^*$ is the imposed displacement, in blue. The symmetry planes represented in green in (a) imply the restricted displacements in (b), depicted in red.
  • Figure 2: 1D linear, elastic problem comprised of $\textcolor{black}{e=1,...,} n_s$ elements of same length $\ell$ and different axial stiffness $(EA)_{e}$ arranged in series.