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.

The time domain boundary element method (TDBEM) is suitable for dealing with dynamic problems in the semi-infinite domain (such as the propagation of seismic waves). The fundamental solution in the TDBEM is an impulse function with time terms, and the formed coefficient matrix is sparse. In this paper, a TDBEM using the compressed storage algorithm based on Compressed Sparse Row (CSR) format is proposed to solve the semi-infinite domain dynamics problems. The coefficient matrix is stored in the CSR format, and the generation method of the coefficient matrix elements and the matrix operation scheme based on CSR format in the TDBEM are given. The GMRES algorithm is also modified to make it suitable for TDBEM of CSR format to improve the efficiency of iterative solution. The provided semi-infinite domain dynamics examples show that the TDBEM of CSR format proposed in this paper can greatly reduce the storage space and improve the efficiency and scale of the solution.