An Integral Method for Solving Dynamic Equations with Fluid–Solid Coupling

In this work, a new methodology is presented to mainly solve the fluid–solid interaction (FSI) equation. This methodology combines the advantages of the Newmark precise integral method (NPIM) and the dual neural network (DNN) method. The NPIM is employed to modify the exponential matrix and loading vector based on the DNN integral method. This involves incorporating the basic assumption of the Newmark-β method into the dynamic equation and eliminating the acceleration term from the dynamic equilibrium equation. As a result, the equation is reduced to a first-order linear equation system. Subsequently, the PIM is applied to integrate the system step by step within the NPIM. The DNN method is adopted to solve the inhomogeneous term through fitting the integrand and the original function with a pair of neural networks, and the integral term is solved using the Newton–Leibniz formula. Numerical examples demonstrate that the proposed methodology significantly improves computing efficiency and provides sufficient precision compared to the DNN method. This is particularly evident when analyzing large-scale structures under blast loading conditions.