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A 3D Bayesian Graph Neural Network Framework for Multiscale Problems

par Petit Valentin - publié le

Vendredi 8 avril 2022
Vasilis Krokos
(Cardiff School of Engineering, Cardiff University, UK)

Multi-scale structures are very common in mechanical and bio-mechanical applications, for instance composite and porous materials. Full Finite Element Analysis (FEA) for stress prediction is usually very expensive in these structures as the FE mesh needs to be very dense to capture the effect of the fine scale features.
In this work we propose a Neural Network (NN) based domain decomposition method to tackle 3D multiscale problems. A wide range of NNs have recently been used to learn the response of FE models in computational mechanics problems. NNs with fully connected layers have been used for instance in (F.Roewer-Desprès et al, 15th International Symposium on Computer Methods in Biomechanics, 2018) but most commonly Convolutional Neural Networks (CNNs) are employed that are very efficient in working with images (V.Krokos et al., Arxiv, 2021). Unfortunately, the computational cost involved in training a CNN for 3D multiscale problems where the resolutions has to be very fine to reflect the effect of the microscale features is prohibitive.
To overcome this problem, we choose to train a Graph NN (GNN) that can operate
directly on the FE mesh used to create the training data. In contrast to images that have a fixed resolution the FE mesh can be much denser in areas where the geometry is complicated or we expect big gradients of our quantity of interest and much coarser everywhere else (T.Pfaff et al., Arxiv, 2021). We use the formulation from (P.W.Battglia, Arxiv, 2018) that allows for a very natural encoding of information on the mesh where absolute information can be encoded on the vertices of the mesh while relative information is encoded on the edges of the mesh.
Lastly, we opt for a Bayesian GNN that is able to quantify the uncertainty of the
prediction and provide not only a mean estimation but also credible intervals for the prediction.