Efficient BEM for thin-walled inhomogeneous potential problems: Theory and MATLAB code

The traditional boundary element method (BEM) often faces challenges in efficiently solving inhomogeneous problems, particularly in thin-walled geometries, due to the need for domain discretization and the handling of nearly singular integrals. In this study, we propose an efficient hybrid algorithm that combines the BEM with physics-informed neural networks (PINNs) to solve inhomogeneous potential problems in thin-walled structures. The approach transforms inhomogeneous equations into equivalent homogeneous ones by subtracting a closed-form particular solution, which is derived using the learning capabilities of PINNs. This methodology not only simplifies the problem formulation but also enhances computational efficiency by eliminating the need for domain discretization, making it particularly well-suited for thin-walled geometries. Additionally, the scaled coordinate transformation BEM, a recently developed technique for solving domain integrals, is also employed for comparative analysis. Finally, a nonlinear coordinate transformation is employed to effectively regularize nearly singular integrals, which are critical in BEM for thin structures. The proposed method achieves accurate and reliable results with a small number of boundary elements, even for structures with extremely small thickness-to-length ratios, as low as 10⁻⁹. This makes the method highly suitable for modeling thin films and thin-walled structures, particularly in the context of advanced smart materials. The unique contribution of this work lies in the integration of PINNs with BEM to tackle challenges specific to thin-walled inhomogeneous problems, offering a more efficient and accurate solution compared to traditional BEM-based method.