Physics-Informed Surrogate Modelling for Dissimilar FSW

Process–Structure–Property Design Space via Gaussian Process Regression

Note: This notebook presents the methodology and analytical framework for an ongoing research project. All numerical demonstrations use representative FSW trial data to match the physical characteristics of the problem.

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📌 Overview

This project develops a physics-informed, uncertainty-aware GPR surrogate model for dissimilar material Friction Stir Welding (FSW). Rather than treating the process as a black box, the work encodes established physical mechanisms directly into the modelling pipeline.

The central hypothesis is the Process → Structure → Property (PSP) linkage:

Process Parameters          →    Microstructure         →    Mechanical Property
(ω, v, plunge depth)             IMC layer thickness          Joint shear strength
     ↕                                  ↕                               ↕
Heat input: Q ∝ ω²/v          Arrhenius diffusion           UTS (MPa)

The analysis demonstrates that Bayesian Optimisation, had it been applied from the start, could have identified the optimal process window in significantly fewer experimental trials than a traditional trial-and-error approach.

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🧠 The Physics Behind It

Heat Input as the Governing Variable

The FSW thermal model (Arbegast & Hartley, 1998) relates peak temperature to process parameters:

$$\frac{T}{T_m} = C \left(\frac{\omega^2}{v \cdot 10^4}\right)^{\alpha}$$

where $T_m$ is the melting point of the softer material (Al), and empirical constants $C \in [0.65, 0.75]$, $\alpha \in [0.04, 0.06]$.

Why the Log Transform Is Not Just a Trick

Intermetallic compound (IMC) growth at the Al–Steel interface follows Arrhenius diffusion kinetics:

$$\delta_{\text{IMC}} \propto D_0 \exp\left(\frac{-E_a}{RT}\right) \cdot \sqrt{t}$$

Since temperature scales with $\ln(\omega^2/v)$, IMC thickness — and by extension joint strength — is approximately linear in log-space. This physical motivation is confirmed empirically: LogHeatInput consistently shows stronger Pearson correlation with UTS than raw HeatInput or individual parameters alone.

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📂 Repository Structure

.
├── FSW_GPR_DesignSpace.ipynb    # Main analysis notebook
├── README.md
└── figures/                     # Auto-generated output figures
    ├── 01_EDA.png
    ├── 02_LOOCV_analysis.png
    ├── 03_DesignSpace.png
    ├── 04_BayesianOptimisation.png
    └── 05_PSP_Chain.png

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🗂️ Notebook Structure

#	Section	Description
1	Background & Motivation	PSP linkage and research rationale
2	Physical Properties Correlation	Arbegast–Hartley model, Arrhenius kinetics
3	Data Loading	Plug-in for real .xlsx data or representative synthetic dataset
4	Exploratory Data Analysis	Correlation maps, design space coverage, heat input distribution
5	Physics-Informed Feature Engineering	Log heat input, RPM/traverse ratio — each with physical justification
6	GPR Modelling	Matérn ν=2.5 kernel, WhiteKernel noise, marginal likelihood optimisation
7	LOOCV Validation	Leave-One-Out CV suited for small-n datasets; RMSE vs noise floor
8	Design Space Characterisation	Probabilistic map: $P(\text{UTS} \geq \tau^*)$ over the full parameter grid
9	Bayesian Optimisation	Expected Improvement acquisition; convergence vs trial-and-error
10	PSP Chain Validation	Physical consistency check against IMC, temperature, force, hardness, EBSD
11	Discussion & Next Steps	Limitations, extensions, broader applicability

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🔬 Modelling Approach

Gaussian Process Regression

GPR was selected over Response Surface Methodology (RSM) for three reasons:

• Non-parametric — no fixed polynomial form assumed; the response surface is learned from data
• Uncertainty-aware — every prediction comes with a calibrated confidence interval, essential when $n < 50$
• Physics-compatible kernel — the Matérn $\nu = 2.5$ kernel produces twice-differentiable functions, appropriate for physical processes that are smooth but not infinitely so

The kernel is defined as:

$$k(\mathbf{x}, \mathbf{x}') = \sigma^2 \cdot \left(1 + \frac{\sqrt{5}r}{l} + \frac{5r^2}{3l^2}\right)\exp\left(-\frac{\sqrt{5}r}{l}\right) + \sigma_n^2 ,\delta(\mathbf{x}, \mathbf{x}')$$

where $r$ is the Euclidean distance between input points, $l$ is the learned length scale, $\sigma^2$ the signal variance, and $\sigma_n^2$ the noise variance absorbed by the WhiteKernel.

Design Space Definition

Following ICH Q8 pharmaceutical analogy, the probabilistic design space is formally defined as:

$$\mathcal{DS}_{p^_} = \left{(\omega, v, d) : P!\left(\tau(\omega, v, d) \geq \tau^_\right) \geq p*\right}$$

where $\tau^*$ is the target shear strength and the probability is evaluated from the GPR predictive distribution $\mathcal{N}(\mu(\mathbf{x}),, \sigma^2(\mathbf{x}))$.

Bayesian Optimisation

The Expected Improvement (EI) acquisition function drives the sequential experiment selection:

$$\text{EI}(\mathbf{x}) = (\mu(\mathbf{x}) - y^* - \xi),\Phi(Z) + \sigma(\mathbf{x}),\phi(Z), \quad Z = \frac{\mu(\mathbf{x}) - y^* - \xi}{\sigma(\mathbf{x})}$$

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⚙️ Getting Started

Prerequisites

pip install numpy pandas matplotlib seaborn scikit-learn scipy openpyxl

Running on Google Colab

The notebook is designed for Colab. Just open it and run all cells — it will mount your Google Drive if you supply your own data, or run on the built-in representative dataset otherwise.

Using Your Own Data

In Section 3, set the flag and file path:

Exp_data = True
FILE_PATH = 'your_data.xlsx'

Your Excel file should contain columns for rotation speed (RPM), traverse speed (mm/min), plunge depth (mm), and measured shear/UTS strength (MPa). Column names are mapped at the top of that cell — adjust them to match your file headers.

Adjusting the Design Space Target

In Section 8, tune the threshold to your application:

UTS_THRESHOLD = 45.0  # MPa — change to your minimum acceptable strength
CONFIDENCE    = 0.90  # 90% probability level for the design space boundary
PLUNGE_FIXED  = 3.0   # mm — fix plunge depth for the 2D parameter map

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📊 Output Figures

Figure	Description
01_EDA.png	Six-panel EDA: log(heat input) vs UTS, RPM–traverse interaction, strength distribution, design space coverage, heat input per trial, plunge depth effect
02_LOOCV_analysis.png	Predicted vs true with 95% CI, residuals vs heat input, per-trial uncertainty
03_DesignSpace.png	Mean UTS surface, prediction uncertainty map, probabilistic design space with 80/90/95% contours
04_BayesianOptimisation.png	EI acquisition per BO step, optimisation convergence vs true optimum
05_PSP_Chain.png	Process → Structure (heat input vs IMC), Structure → Property (IMC vs UTS), GPR uncertainty vs IMC

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📐 Feature Engineering Summary

Feature	Formula	Physical Basis
Heat Input Proxy	$Q = \omega^2 / v$	Arbegast & Hartley (1998) FSW thermal model
Log Heat Input	$\ln(\omega^2 / v)$	Linearises Arrhenius IMC growth kinetics — primary model input
RPM/Traverse Ratio	$\omega / v$	Simplified heat proxy; easier to interpret for process engineers

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📖 References

• Arbegast, W.J. & Hartley, P.J. (1998). FSW Technology. TMS.
• Rasmussen, C.E. & Williams, C.K.I. (2006). Gaussian Processes for Machine Learning. MIT Press.
• Arlot, S. & Celisse, A. (2010). A survey of cross-validation procedures for model selection. Statistics Surveys, 4, 40–79.
• Shahriari, B. et al. (2016). Taking the human out of the loop: A review of Bayesian optimization. Proceedings of the IEEE, 104(1), 148–175.
• Shi, S. et al. (2023). Gaussian process regression for materials property prediction. npj Computational Materials.

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🤝 Contributing

Issues and suggestions are welcome — particularly around kernel selection, alternative acquisition functions for BO, or extension to 3D design space visualisation with plunge depth as a free variable.

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Made with 🔥 and a lot of thermocouple data