4.6 Article

Adaptive sinh transformation Gaussian quadrature for 2D potential problems using deep learning

Journal

ENGINEERING ANALYSIS WITH BOUNDARY ELEMENTS
Volume 155, Issue -, Pages 197-211

Publisher

ELSEVIER SCI LTD
DOI: 10.1016/j.enganabound.2023.06.002

Keywords

Boundary element method; Nearly singular integrals; Sinh transformation method; Deep learning

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A novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), is proposed based on deep learning, which can determine the number of Gaussian points according to the required accuracy. Compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency.
In the boundary element method (BEM), the sinh transformation method is an effective method for evaluating nearly singular integrals, but a relationship between the integration accuracy and the number of Gaussian points is needed to achieve adaptive computation. Based on deep learning, we propose a novel integration scheme, adaptive sinh transformation Gaussian quadrature (ASTGQ), which can determine the number of Gaussian points according to the required accuracy. First, a large number of integration data samples of the sinh transformation method are generated in different cases, and the neural network is trained to establish the relationship between the number of Gaussian points and the integration accuracy. Then, based on the improved loss function and evaluation index, a better network model is obtained to ensure that the actual integration accuracy is slightly higher than the requirement of using the minimum Gaussian points. In this way, when the trained neural network is used in the sinh transformation method, the higher accuracy requirement can be met at a lower cost. Numerical examples demonstrate that, compared to the adaptive Gaussian quadrature (AGQ) method, the proposed scheme can significantly improve the computational efficiency when evaluating the nearly singular integrals for very thin coatings and other structures.

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