Multi-objective Bayesian Optimisation of Spinodoid Cellular Structures for Crush Energy Absorption

TL;DR

This paper presents a multi-objective Bayesian optimisation framework for designing spinodoid metamaterials aimed at crash energy absorption, balancing energy absorption and peak force under complex, non-linear material behaviour. The approach couples finite-element simulations (FEM) with Gaussian-process surrogates and leverages both scalarisation (ParEGO, weighted sum) and hypervolume (EHVI) MOBO strategies, enhanced by a gradient-based filter to avoid densification. Material anisotropy is modelled with a Hill-type constitutive law calibrated from PET-G experiments, and spinodoid topology is generated via GRF-based anisotropic topologies with parameters $\Theta=\{\rho,\lambda,\theta_1,\theta_2,\theta_3\}$. Key findings show that MOBO, particularly qNEHVI, can efficiently identify a diverse set of Pareto-optimal designs (e.g., 78 points) that maximize energy absorption while curbing peak forces, with notable reductions in PF achieved without sacrificing EA; a three-variable, fixed-density study further accelerates optimization and yields bending-dominated, bone-like topologies. The framework demonstrates a scalable, data-efficient route to crash-worthy designs and is extensible to multi-physics optimisation and multi-material problems in engineering.

Abstract

In the pursuit of designing safer and more efficient energy-absorbing structures, engineers must tackle the challenge of improving crush performance while balancing multiple conflicting objectives, such as maximising energy absorption and minimising peak impact forces. Accurately simulating real-world conditions necessitates the use of complex material models to replicate the non-linear behaviour of materials under impact, which comes at a significant computational cost. This study addresses these challenges by introducing a multi-objective Bayesian optimisation framework specifically developed to optimise spinodoid structures for crush energy absorption. Spinodoid structures, characterised by their scalable, non-periodic topologies and efficient stress distribution, offer a promising direction for advanced structural design. However, optimising design parameters to enhance crush performance is far from straightforward, particularly under realistic conditions. Conventional optimisation methods, although effective, often require a large number of costly simulations to identify suitable solutions, making the process both time-consuming and resource intensive. In this context, multi-objective Bayesian optimisation provides a clear advantage by intelligently navigating the design space, learning from each evaluation to reduce the number of simulations required, and efficiently addressing the complexities of non-linear material behaviour. By integrating finite element analysis with Bayesian optimisation, the framework developed in this study tackles the dual challenge of improving energy absorption and reducing peak force, particularly in scenarios where plastic deformation plays a critical role. The use of scalarisation and hypervolume-based techniques enables the identification of Pareto-optimal solutions, balancing these conflicting objectives.

Figures (31)

• Figure 1: Visualisation of changes in two design parameters using graded spinodoids. Both structures were produced with $\theta_1 = 90^\circ, \theta_2 = 0^\circ, \theta_3 = 0^\circ$. (a) A linearly graded generated with an increasing relative density from left to right. The left side has the lowest relative density, starting at 0.3, the middle portion has a relative density of 0.45, and the right side reaches a relative density of 0.6. (b) A structure illustrating an increase in the wave number from left to right. The left side has the lowest wave number of $4\pi$, the right has the highest at $20\pi$, and the middle portion has an intermediate value of $12\pi$. It should be noted that these structures serve as a visualisation tool and do not represent the structures being optimised.
• Figure 2: Four unique topologies generated through a unique choice of $\theta$, serving as benchmarks for FEM validation and showcasing the robustness of the design space. The topologies were generated assuming $\rho = 0.3$, and $\lambda = 15\pi$. (a) The anisotropic 'Columnar' topology features uniformly distributed, interconnected column-like structures oriented along the direction of loading. (b) The 'Isotropic' topology is produced with relatively high conical angles. (c) The 'Cubic' structure arises from prescribing relatively small conical angles of equal magnitude. (d) The 'Lamellar' structure exhibits anisotropy in the direction perpendicular to the loading direction.
• Figure 3: Results of material characterisation tests for Polyethylene terephthalate glycol (PET-G) under compression and tension are presented. Negative values indicate compression, while positive values indicate tension. The solid red line, dashed blue line, and dash-dotted green line represent specimens printed at $0^\circ$, $45^\circ$, and $90^\circ$, respectively. The subfigures in the top left illustrate the general printing directions used for manufacturing the test samples, with black arrows indicating the printing angle from the horizontal and blue arrows showing the loading directions.
• Figure 4: Comparison of stress-strain behaviour of structures generated using the conical angles described in Fig. \ref{['fig:4 spinodoid types']}, undergoing crushing with a relative density of $\rho = 0.3$ and a higher wave number of $\lambda = 24\pi$. The results were obtained from both experiments and FEM analysis. The three different black lines represent results from each repeated experiment, while the solid red line represents data from FEM analysis. The subfigures above each plot show a visual comparison between the 3D-printed structures (left) and the contour plots from FEM analysis (right).
• Figure 5: The Force-Displacement (F-D) diagram illustrates the typical behaviour of cellular structures under compressive loading, which can be divided into three stages: i) The pre-crushing stage involves linear elastic deformation occurring until peak force has been attained, followed by ii) the force plateauing as a result of a combination of plastic deformation, and fracture, this is called the progressive crushing zone, iii) the final phase involves the structure undergoing densification signified by an exponential increase in the crushing load required for further deformation.