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Title: | Gradient-Based Calibration of a Precipitation Hardening Model for 6xxx Series Aluminium Alloys |
Authors: | Alizadeh, A Souissi, M Zhou, M Assadi, H |
Keywords: | precipitation hardening;gradient-based optimisation;model calibration;6xxx series alloys;physics-informed machine learning (PIML);physics-based modelling (PBM) |
Issue Date: | 19-Sep-2025 |
Publisher: | MDPI |
Citation: | Alizadeh, A. et al. (2025) 'Gradient-Based Calibration of a Precipitation Hardening Model for 6xxx Series Aluminium Alloys', Metals, 15 (9), 1035, pp. 1 - 25. doi: 10.3390/met15091035. |
Abstract: | Precipitation hardening is the primary mechanism for strengthening 6xxx series aluminium alloys. The characteristics of the precipitates play a crucial role in determining the mechanical properties. In particular, predicting yield strength (YS) based on microstructure is experimentally complex and costly because its key variables, such as precipitate radius, spacing, and volume fraction (VF), are difficult to measure. Physics-based models have emerged to tackle these complications utilising advancements in simulation environments. Nevertheless, pure physics-based models require numerous free parameters and ongoing debates over governing equations. Conversely, purely data-driven models struggle with insufficient datasets and physical interpretability. Moreover, the complex dynamics between internal model variables has led both approaches to adopt heuristic optimisation methods, such as the Powell or Nelder–Mead methods, which fail to exploit valuable gradient information. To overcome these issues, we propose a gradient-based optimisation for the Kampmann–Wagner Numerical (KWN) model, incorporating CALPHAD (CALculation of PHAse Diagrams) and a strength model. Our modifications include facilitating differentiability via smoothed approximations of conditional logic, optimising non-linear combinations of free parameters, and reducing computational complexity through a single size-class assumption. Model calibration is guided by a mean squared error (MSE) loss function that aligns the YS predictions with interpolated experimental data using L2 regularisation for penalising deviations from a purely physics-based modelling structure. A comparison shows that the gradient-based adaptive moment estimation (ADAM) outperforms the gradient-free Powell and Nelder–Mead methods by converging faster, requiring fewer evaluations, and yielding more physically plausible parameters, highlighting the importance of calibration techniques in the modelling of 6xxx series precipitation hardening. |
Description: | Data Availability Statement:
The original contributions presented in this study are included in the article/Supplementary Material . Further inquiries can be directed to the corresponding authors. Supplementary Materials: The following supporting information can be downloaded at: https://www.mdpi.com/article/10.3390/met15091035/s1. GIF S1: Gradient-based calibration of the predicted yield strength curve using the ADAM optimiser. The model iteratively adjusts parameters to minimise the loss, with a stopping criterion set to a function tolerance of 1 × 10^−6. The convergence towards the interpolated experimental data is shown across successive iterations. |
URI: | https://bura.brunel.ac.uk/handle/2438/32061 |
DOI: | https://doi.org/10.3390/met15091035 |
Other Identifiers: | ORCiD: Amir Alizadeh https://orcid.org/0000-0002-2319-8596 ORCiD: Maaouia Souissi https://orcid.org/0000-0002-8451-7909 ORCiD: Mian Zhou https://orcid.org/0000-0002-6256-8676 ORCiD: Hamid Assadi https://orcid.org/0000-0001-5327-1793 Article number: 1035 |
Appears in Collections: | Brunel Centre for Advanced Solidification Technology (BCAST) |
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