Isogeometric boundary element analysis of nonlinear liquid sloshing in containers under pitching oscillation

Abstract

This paper proposes an isogeometric boundary element method (IGABEM) to solve the nonlinear liquid sloshing problem in a rectangular container subjected to oscillatory excitation. Based on the semi-Lagrange approach, a fixed global coordinate system and a local Cartesian coordinate system that moves synchronously with the container are defined. Starting from the Laplace equation, the boundary integral equations for the liquid sloshing problem are derived using Gauss’s divergence theorem and the integration by parts technique, while incorporating nonlinear kinematic and dynamic boundary conditions of the free surface. The corresponding boundary element solution system is then formulated. Non-Uniform Rational B-Splines (NURBS) are employed as shape functions to accurately describe the geometric boundaries and approximate the unknown physical fields. This method ultimately produces the discrete equations governing nonlinear liquid sloshing problem in an oscillating container. Compared with traditional polynomial interpolation shape functions, NURBS provide improved continuity both within elements and across element interfaces as well as local support. These properties make them particularly suitable for satisfying the continuity requirements of the liquid surface. For time integration, a second-order Runge–Kutta algorithm is employed for time-stepping to solve the IGABEM system equations, compute variable gradients at each time step, and update the computational grid in real-time. A series of numerical examples are presented, the results are compared with analytical solutions, experimental data, and alternative numerical methods for free and forced liquid sloshing, free surface fluctuations and internal pressures. These comparisons validate the accuracy and robustness of the proposed method. The numerical examples further investigate the effects of external excitation frequency, excitation amplitude, rotation center position, and bottom obstacle height on liquid sloshing responses in the rectangular container. The results indicate that changes in excitation frequency, vertical eccentricity of the rotation center, and obstacle height significantly influence the liquid sloshing behavior.