Structural vibrations are a source of unwanted noise in engineering systems like cars, trains or airplanes. Minimizing these vibrations is crucial for improving passenger comfort. This work presents a novel design optimization approach based on guided flow matching for reducing vibrations by placing beadings (indentations) in plate-like structures. Our method integrates a generative flow matching model and a surrogate model trained to predict structural vibrations. During the generation process, the flow matching model pushes towards manufacturability while the surrogate model pushes to low-vibration solutions. The flow matching model and its training data implicitly define the design space, enabling a broader exploration of potential solutions as no optimization of manually-defined design parameters is required. We apply our method to a range of differentiable optimization objectives, including direct optimization of specific eigenfrequencies through careful construction of the objective function. Results demonstrate that our method generates diverse and manufacturable plate designs with reduced structural vibrations compared to designs from random search, a criterion-based design heuristic and genetic optimization.
Preprint available from arxiv.
This repository mainly contains the following elements:
- (Unguided) flow matching code to generate novel beading patterns in
plate_optim/flow_matching. - Surrogate / regression code to predict dynamic responses of plates with beading patterns in
plate_optim/regression. - A combination of the regression resulting in a guided flow matching solver in
plate_optim/guided_flow.pyandplate_optim/utils/guidance.py. - Loss functions to guide towards low vibrations in
plate_optim/utils/guidance.py. - Metrics to assess the manufacturability of generated plate designs in
plate_optim/metrics. - A procedural beading pattern design generation method, with no direct relation to vibrations, in
plate_optim/pattern_generation.py. - A workflow to evaluate beading patterns via FE computation
notebooks/simulate_plate_via_fem.ipynb
If you would like to try out our method, we recommend getting started with the notebook saved in notebooks/run_guided_fm_design_optimization.ipynb
Our dataset consisting of beading patterns and associated numerical simulations is available from here.
The individual files can be downloaded via e.g. wget:
- D500_edit.h5 https://data.goettingen-research-online.de/api/access/datafile/125125
- D50k_edited.h5 https://data.goettingen-research-online.de/api/access/datafile/125180
- moments_D50k.pt https://data.goettingen-research-online.de/api/access/datafile/125124
- flow_matching_0331.ckpt https://data.goettingen-research-online.de/api/access/datafile/125254
- regression_075_noise_0312.ckpt https://data.goettingen-research-online.de/api/access/datafile/125255
- regression_no_noise_0526.ckpt https://data.goettingen-research-online.de/api/access/datafile/125253
Our numerical simulations are performed with elpaso, which is a fast and flexible simulation tool developed specifically for vibroacoustics. Elpaso is not provided as part of this package. The numerical simulations can also be performed with other finite-element solvers. As an example, we provide directions how to evaluate the plate model with Abaqus notebooks/simulate_plate_via_fem.ipynb.
The video is an example of the optimization process for a beading pattern. It shows the beading pattern starting from random noise, the gradient signal from the regression model and the prediction from the flow matching model (from left to right). The red cross marks the loading position.
combined_0.mp4
If you would like to try out our design optimization method, we recommend getting started with with the notebook saved in notebooks/run_guided_fm_design_optimization.ipynb
If you would like to train models yourself, first git clone this repository and then set up the environment following the environment.yml file:
conda create -f environment.yml
pip install -e .
To specify where you want to save the data, models, etc., you can change the paths in plate_optim/project_directories.py
This repository employs wandb for logging. To be able to use it you must have an account and login: wandb login
Evaluation: This repository does not contain code for performing numerical simulations. We perform simulations with elpaso. We provide a notebook to evaluate plates with beading patterns using the FEM software Abaqus notebooks/simulate_plate_via_fem.ipynb.
All scripts require that the datasets are downloaded and the paths are correctly set up as described in Getting started.
Run one of the scripts in scripts/regression, e.g.:
source scripts/regression/50k15_075noise.sh
Run the script in scripts/flow_matching:
source scripts/flow_matching/standard.sh
Performing design optimization requires already trained models, which can be downloaded as described in the data section of this readme. To run flow matching for the 100-200 Hz frequency range and simply supported plates, run the following:
source scripts/range100_200.sh
source scripts/setting_a_args.sh
source scripts/experiment_scripts/run_flow_matching.sh
The first two scripts define environmental variables for the frequency range, the boundary condition, the model checkpoints, etc..
The third script actually calls plate_optim/guided_flow.py with the set variables.
Before running these scripts, you probably need to adjust the paths inside setting_a_args.sh to actually point to the model checkpoints
and you might want to change the directory where results are saved in run_flow_matching.sh. Alternatively, the relevant environmental variables can be provided directly in the terminal like this:
min_freq=100
max_freq=200
n_freqs=101
freqs='100_200'
extra_conditions='[0,0.34444444444444444,0.35]' # 0, 0.31 / 0.9, 0.21 / 0.6 # clamping, loading position x, loading position y
flow_matching_path='path'
regression_path_noise='path'
setting='a'
source scripts/experiment_scripts/run_flow_matching.sh
This block gives an overview over the available scripts:
scripts| |____range100_200.sh # specify frequency range 100-200 Hz |____range200_250.sh |____setting_a_args.sh # specify boundary condition setting according to paper |____setting_b_args.sh |____experiment_scripts # after specifying boundary condition and frequency range run one of the optimization methods | |____run_random_search.sh # baseline method | |____run_flow_matching.sh # standard guided flow matching | |____run_fm_first_peak.sh # flow matching for optimizing the first eigenfrequency | |____run_genetic_opt.sh # baseline method | |____further_scripts.sh # other scripts for ablation studies
Most of our code is MIT licensed. The unguided flow matching code in this repository builds mainly upon the flow matching package which is CC BY-NC licensed and therefore has the same license. This license applies to the code within the plate_optim/flow_matching directory. The rest of our repository is released under the MIT license. Please note, that the published data is not licensed under MIT, as detailed here.
@article{delden2025minimizing,
author={van Delden, Jan and Schultz, Julius and Rothe, Sebastian and Libner, Christian and Langer, Sabine C. and L{\"u}ddecke, Timo},
title={Minimizing Structural Vibrations via Guided Flow Matching Design Optimization},
journal={arXiv preprint arXiv:2506.15263},
year={2025},
}
The regression model employed in this work builts upon our previous project on predicting structural vibrations.
@inproceedings{delden2024vibrations,
title={Learning to Predict Structural Vibrations},
author={van Delden, Jan and Schultz, Julius and Blech, Christopher and Langer, Sabine C and L{\"u}ddecke, Timo},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=i4jZ6fCDdy}
}
Our work is in part built upon the following wonderful repositories: