Abstract
This paper applies a machine learning technique to find a general and efficient numerical integration scheme for boundary element methods. A model based on the neural network multi-classification algorithm is constructed to find the minimum number of Gaussian quadrature points satisfying the given accuracy. The constructed model is trained by using a large amount of data calculated in the traditional boundary element method and the optimal network architecture is selected. The two-dimensional potential problem of a circular structure is tested and analyzed based on the determined model, and the accuracy of the model is about 90%. Finally, by incorporating the predicted Gaussian quadrature points into the boundary element analysis, we find that the numerical solution and the analytical solution are in good agreement, which verifies the robustness of the proposed method.
Highlights
The methods for solving partial differential equations (PDEs) are usually classified as analytical and numerical methods
The boundary element method (BEM) is a numerical method in solving partial differential equations (PDEs) which can be formed into boundary integral equations (BIE)
15 Gaussian quadrature points are used in Case 1 to calculate the coefficient matrix, and the minimum number of Gaussian quadrature points obtained by machine learning prediction are used in Case 2 to calculate the coefficient matrix
Summary
The methods for solving partial differential equations (PDEs) are usually classified as analytical and numerical methods. IGA intends to bridge CAD and Computer-Aided Engineering (CAE) by employing the basis functions constructing CAD models to discretize the PDE in numerical simulations. Because both the BEM and CAD are based on boundary representation [5–7], they are naturally compatible with each other. The coefficient matrix of the BEM is an asymmetric full-matrix, so the computational time increases rapidly with the degrees of freedom This process can be accelerated by algorithms such as Fast Multipole Method (FMM), Adaptive Crossing Approximation (ACA), and fast Fourier transformation, etc. The machine-learning enhanced FEM has been applied to investigate the numerical quadrature scheme [32], estimation of stress distribution [33], construction of smart elements [34], data-driven computing paradigm [35], and structural optimization [36].
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