Nonlinear analysis of liquid sloshing in containers under pitching load with scaled boundary finite element method

Abstract

A semi-analytical numerical model based on the scaled boundary finite element method (SBFEM) is proposed for analyzing nonlinear liquid sloshing in containers subjected to pitching excitation. To track the motion of the liquid free surface, the Semi-Lagrangian (SL) method is employed, with two Cartesian coordinate systems comprising a fixed inertial system and a moving system. Meanwhile, a second-order Runge–Kutta algorithm (RK2) ensures accurate temporal updates of the physical variables and their gradients. Within this framework, the governing equation, along with the boundary conditions, is reduced to a second-order ordinary differential equation using the weighted residual method. Then, dual variables are introduced to reduce the order of the equation for solution. Compared to the finite element method, the proposed approach requires only boundary discretization while retaining mesh refinement flexibility. Analytical solution procedures are available in the radial direction, improving accuracy with fewer degrees of freedom along the circumferential direction. Unlike the boundary element method, it eliminates the need for fundamental solutions and avoids singular integrals. The validity and accuracy of the method are verified against other methods by comparing the change in free surface elevation and forces induced by liquid sloshing. The validity and accuracy of the method are further demonstrated by the computational time and the error associated with the computational paradigm. Subsequently, a systematic analysis is conducted to examine the effects of excitation frequencies, filling levels, and vertical eccentric distance on the liquid sloshing behavior. Finally, how the radius of bottom-rounded corners affects the dynamic sloshing force and surface elevation is investigated as well. The results show that the considered factors affect surface elevation, sloshing frequencies, and the dynamic forces acting on containers under pitching excitation to varying degrees, which may provide important guidance and a scientific basis for sloshing reduction in liquid storage structures.