In large-scale hydrodynamic analyses of large floating structures, conventional iterative algorithms frequently encounter ill-conditioning when solving linear systems derived from boundary element discretisation, typically manifesting as significantly degraded convergence rates or numerical instabilities. To address this challenge, we propose a novel Improved Modified Gram-Schmidt GMRES (IMGS-GMRES) algorithm. This algorithm significantly improves the efficiency and numerical stability in solving ill-conditioned systems by optimising the orthogonalisation process and incorporating preconditioning techniques. The method combines preconditioning strategies with an enhanced Gram-Schmidt orthogonalisation technique to effectively mitigate high condition numbers. The IMGS-GMRES algorithm has been implemented within the in-house hydrodynamic solver FHLAN and validated against the commercial software WADAM, demonstrating improvements in computational accuracy and stability. Furthermore, it optimises the solution process for large-scale linear systems encountered in hydrodynamic analysis. Numerical experiments demonstrate superior performance: for a system with a condition number of 1.06×10²², IMGS-GMRES reduces iterations by 44.8 % (from 498 to 275) and suppresses orthogonality error to the machine precision of 4.11×10⁻¹⁶. In a large-scale case with 31,000 panels, it achieves 82 % runtime reduction vs direct solvers. The algorithm provides an efficient, robust solution for ill-conditioned hydrodynamic analyses of large floating bodies and high-dimensional coupled systems, supporting real-time simulation and optimisation.

To address the neglect of real-time mass transfer effects by traditional analysis methods during the side-by-side offloading process of Floating Production Storage and Offloading (FPSO) and shuttle tankers, a numerical model incorporating the variable mass effect is established to enable the simulation of dynamic offloading processes. Using this model, the dynamic response characteristics under different offloading rates and sea conditions are systematically investigated and validated against towing tank tests. Based on the previously optimized benchmark configuration, which includes 16 side-by-side mooring lines, six floating fenders, and an anchor line angle of 60° for the FPSO, the evolution laws of mooring line tension and fender pressure under different offloading rates were systematically investigated under normal and extreme sea conditions. The results show that an increase in offloading rate significantly amplifies the system’s fender load; when the offloading rate reaches approximately 1.4 t/s, the system transitions from the quasi-static response region to a significant nonlinear coupling region, demonstrating obvious sea condition–rate coupling characteristics. Under the combined action of high offloading rates and severe sea conditions, fender pressure rapidly approaches the design limit, becoming the primary safety bottleneck for the system. Model test results indicate that the numerical model can well predict mechanical responses under low and medium offloading rate conditions. The research results can provide a reference for offloading rate control, safety assessment, and operational window determination during FPSO side-by-side dynamic offloading operations.