
Graph Neural Network-based Gauss Quadrature for Isogeometric Analysis
Please login to view abstract download link
Isogeometric analysis (IGA) is a computational method for solving partial differential equations (PDEs). It utilizes non-uniform rational B-splines (NURBS) to define the geometry and approximate the solution field [1]. One crucial aspect of IGA for linear-elastic problems is calculating the element stiffness matrix. The Gauss quadrature rule is commonly employed to perform the numerical integration required to compute the stiffness matrix [1]. The accuracy of the computed element stiffness matrix in IGA depends on the number of Gauss quadrature points (GQPs) used for numerical integration. Deformed elements typically require higher values of Gauss quadrature points to capture the material’s behavior compared to undeformed elements accurately [2]. However, using larger values of the GQPs increases computational costs. Therefore, finding optimal values for the GQPs that balance accuracy and computational efficiency when computing the stiffness matrix is important. Recent developments in computational mechanics have seen a surge in the integration of machine learning approaches, demonstrating a trend toward utilizing advanced computational tools for more effective problem-solving and analysis [3]. One such approach involves the use of graph neural networks (GNNs) [4]. GNNs are a class of deep learning methods designed to extract insights and make predictions from data structured as graphs. In our study, we applied a GNN to predict the values of Gauss quadrature points. Specifically, our GNN model takes second-order NURBS elements as input and predicts the optimal values for the GQPs from the set of {5, 6, 7, 8, 9}. The model’s overall accuracy on the training and testing data are 72% and 71%, respectively. This approach allows us to determine the appropriate quadrature points efficiently for accurately computing the element stiffness matrix in IGA. To validate the model’s performance, we solve two numerical problems: a cantilever beam and an infinite plate with a hole. REFERENCES [1] J. A. Cottrell, T. J. R. Hughes, and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, 2009. [2] A. Oishi and G. Yagawa. Computational mechanics enhanced by deep learning. Computer Methods in Applied Mechanics and Engineering, 327:327-351, 2017. [3] S. Kollmannsberger et al., Deep Learning in Computational Mechanics. 1st Edition, Springer, 2021. [4] Y. Ma and J. Tang. Deep Learning on Graphs. Cambridge University Press, 2021.