Open-source Code for Multi-objective Bayesian Optimisation of Spinodoid Cellular Materials

This repository introduces a multi-objective Bayesian optimisation (MOBO) framework for optimising spinodoid structures—scalable, non-periodic topologies with efficient stress distribution—to enhance crush energy absorption under impact. The framework addresses the challenge of balancing conflicting objectives: maximising energy absorption while minimising peak forces, accounting for non-linear material behavior and plastic deformation. By integrating finite element analysis (FEA) with Bayesian optimisation, it efficiently navigates the design space, reducing computational costs compared to conventional methods (e.g., NSGA-II). Key features include:

• Pareto-optimal solutions via scalarisation and hypervolume techniques.

• Avoidance of structural densification to maintain integrity.

• Superior performance over NSGA-II in computational efficiency and solution quality.

Ideal for real-world structural/material optimisation where complex trade-offs and non-linear dynamics are critical.

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: A schematic of the multi-objective Bayesian optimisation (MOBO) process for optimising the spinodoid cellular structure design space involves four steps of the data-driven process. (1) Initial Dataset Creation: Sampling 50 points from the design space using the Sobol' sequence and evaluate them with FEM simulations to build the initial dataset. (2) Surrogate Model Update: Updating the Gaussian Process model based on the dataset to predict structural properties. (3) Identifying samples to evaluate: Using an acquisition function to identify and evaluate the most promising design points. (4) FEM Simulations: Performing FEM analysis on generated structures and extracting objectives to then update the dataset.

Figure 3: Optimisation history: peak force verus energy absorption. Insert figure: the hypervolume evoluation against iterations.

Figure 4: Topologies changes with the changing pareto front.

MOBO is an open-source framework capable of carrying out Bayesian optimisation of spinodoid cellular materials.

First author: Hirak Kansara, Code contribution: Siamak Khosroshahi, Corresponding author: Dr Wei Tan (wei.tan@qmul.ac.uk)

• Python 3.8+
• pip
• FEA software (Abaqus)
• MATLAB (Requires the GIBBON library https://github.com/gibbonCode/GIBBON.git and ImageProcessingToolBox)

1. Clone the repository:

   git clone https://github.com/MCM-QMUL/MOBO.git
   cd MOBO 

2. Install dependencies:

   pip install -r requirements.txt  

   Key packages include:

   • numpy, scipy: Numerical operations.
   • botorch, gpytorch: Bayesian optimisation.
   • pymoo: Multi-objective optimisation utilities.

3. Change directories in:

   • config.yaml to define path to MOBO folder, path to spinodal resources, and path to GIBBON folder
   • MOBO_standalone\spinodal_resources\spinodoid_scripy.py to allow subprocess to call MATLAB.exe file
   • Add path to GIBBON folder in Objective_Spinodoid_Tet.m lines 6 & 7

Run the optimisation workflow with:

python main.py

• input_columns: Design parameters.
• n_iterations: Number of MOBO iterations used as stopping criteria
• kernel: Type of kernel used for covariance calculation
• MOBO_type: MOBO methods as described in the article
• apply_transform: If True, scales the objectives for them to be maximised.

• output_columns: Parameters to output, maximised by default
• Pareto-optimal designs: Saved to hydra_output_dir/pareto_front.csv.
• Optimisation History: Saved to hydra_output_dir/optimisation_history.csv.
• Simulation logs: Stored in hydra_output_dir/main.log.

1. Define objectives in config.yaml:

   objectives:  
     EA 
     Peak_Force

2. Start optimisation: # change mode in main.py if running framework either locally or on cluster

   python main.py  

3. Analyse results:

   • View Pareto front: hydra_output_dir/pareto_front.csv.

• Hardware: Simulations are computationally intensive. Use HPC/cluster for large-scale runs.
• Troubleshooting:
  • If FEA fails, check <job_name>.o<job_id>.
  • Mesh density can be changed in /MOBO_standalone/spinodal_resources/Objective_Spinodoid_Tet.m by changing the res variable for faster debugging.

If using this code for research or industrial purposes, please cite:

[1] Kansara, H., Khosroshahi, S.F., Guo, L., Bessa, M.A. and Tan, W., Multi-objective Bayesian Optimisation of Spinodoid Cellular Structures for Crush Energy Absorption. Computer Methods for Applied Mechanics and Engineering, 2025. https://doi.org/10.1016/j.cma.2025.117890

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