surrogate-shm-model

Author: Mahdi Mohammadzadeh

A data-driven surrogate model that replaces slow FEM simulation of a cantilever beam with a trained ML model — achieving millisecond inference at FEM-level accuracy.

Overview

This project builds a complete surrogate modeling pipeline for structural simulation. A cantilever beam under a point load is modeled with Euler-Bernoulli beam theory, generating 10,000 labeled samples across realistic engineering ranges. Two machine learning surrogates are then trained to predict the beam response directly from geometry, material, and load parameters.

The goal is to replace repeated slow FEM-style simulation calls with a trained model that can estimate structural response in milliseconds while preserving engineering-level accuracy.

What the model predicts

The surrogate receives six input features:

Parameter	Symbol	Range	Unit
Beam length	L	0.5 to 3.0	m
Cross-section width	b	0.01 to 0.10	m
Cross-section height	h	0.01 to 0.10	m
Young's modulus	E	70e9 to 210e9	Pa
Applied load	F	100 to 10000	N
Density	rho	2700 to 7850	kg/m^3

It predicts three structural responses:

Response	Symbol	Unit
Tip deflection	delta	m
Max bending stress	sigma	Pa
First natural frequency	f1	Hz

Physics model

Euler-Bernoulli beam theory is used as the fast analytical simulator for a cantilever beam with a point load at the free end.

Quantity	Equation	Description
I	I = (b * h^3) / 12	Second moment of area, m^4
A	A = b * h	Cross-sectional area, m^2
m	m = rho * A * L	Total beam mass, kg
delta	delta = (F * L^3) / (3 * E * I)	Tip deflection, m
sigma	sigma = (F * L * (h / 2)) / I	Max bending stress, Pa
f1	f1 = (1 / (2 * pi)) * sqrt(3 * E * I / (m * L^3))	First natural frequency, Hz

Dataset

The notebook generates data/beam_simulation_data.csv with 10,000 rows and 9 columns: six input parameters and three target responses. All random seeds are fixed at 42 for reproducibility.

The input distributions are uniform by construction, which gives broad coverage of the design space before model training.

The target distributions are nonlinear because deflection, stress, and frequency depend on powers of beam length and cross-section height.

The correlation heatmap gives a quick diagnostic view of feature-target relationships. It also shows why a nonlinear model is useful: some important relationships are not captured by simple linear correlation.

Models

Two surrogate models are trained in the notebook:

Model	Implementation	Notes
XGBoost	MultiOutputRegressor(XGBRegressor)	Tree-based nonlinear regressor trained on log-transformed targets
MLP	PyTorch SurrogateNet	Three hidden layers: 128, 128, 64 with ReLU and batch normalization

The saved production artifact is:

models/xgb_surrogate.pkl

Results

The executed notebook reports the following test-set performance:

Target	XGBoost R2	MLP R2	Best model
delta	0.910040	0.965500	MLP
sigma	0.984725	0.976768	XGBoost
f1	0.994884	0.989792	XGBoost

All three target responses exceed R2 > 0.95 in at least one surrogate model.

The MLP loss curve shows stable convergence over 200 epochs.

Inference benchmark

The benchmark compares vectorized analytical evaluation against surrogate inference on the full test set of 2,000 samples.

Method	Samples	Time (ms)	Throughput (samples/sec)
Simulator	2000	0.616400	3,244,646.29
XGBoost	2000	8.786200	227,629.69
MLP	2000	1.508800	1,325,556.73

The analytical simulator here is a vectorized closed-form calculation, so it is not representative of full FEM runtime. In a practical FEM workflow, the trained surrogate replaces expensive simulation solves with millisecond-scale model inference.

Repository layout

surrogate-shm-model/
├── data/
│   └── beam_simulation_data.csv
├── docs/
│   └── figures/
├── models/
│   └── xgb_surrogate.pkl
├── notebooks/
│   └── surrogate_model.ipynb
├── src/
│   └── beam_simulator.py
├── README.md
└── requirements.txt

Reproduce

Install dependencies:

pip install -r requirements.txt

Run the executed notebook pipeline:

jupyter nbconvert --to notebook --execute notebooks/surrogate_model.ipynb --output surrogate_model.ipynb --ExecutePreprocessor.timeout=300

Stack

Python, NumPy, Pandas, scikit-learn, XGBoost, PyTorch, Matplotlib, Seaborn, Joblib, Jupyter.