Semi-analytical analysis of nonlinear liquid sloshing in rectangular tanks with scaled boundary finite element method

Abstract

This study presents a novel semi-analytical model for nonlinear liquid sloshing response of two-dimensional (2D) liquid storage tanks in the context of the scaled boundary finite element method (SBFEM). The potential flow in the tank is governed by the 2D Laplace equation, with the free surface considered as a nonlinear boundary condition. To trace the motions of the liquid-free surface, the semi-Lagrange (SL) method is employed, and two Cartesian coordinate systems are established, including a fixed inertial system and a moving system. Meanwhile, a fourth-order Runge–Kutta (RK4) algorithm is employed for achieving updates of the physical variables and their gradients. A scaled boundary coordinate system is established, encompassing circumferential and radial directions. Within this framework, the SBFEM equation in form of second-order ordinary differential equation is derived by using the weighted residual method. Subsequently, a dual variable comprising nodal potential and flux, along with an eigenfunction expansion method, is introduced into the solution procedure. The proposed approach combines the strengths of both boundary element and finite element methods, requiring only boundary discretization for numerical simulation, thus reducing the spatial dimension by one, and the solution possesses analytical properties in the radial direction. Importantly, the proposed SBFEM model does not require a fundamental solution, eliminating the need for treating singular integrations, as is common in traditional boundary element method. Numerical examples confirm the superior computational accuracy, convergence rate, and efficiency of our method compared to other numerical approaches. The method exhibits insensitivity to the time step selection and the computational accuracy can be further improved by increasing grid density or element order. Moreover, numerical experiments on U-shaped aqueducts demonstrate its applicability to analyzing nonlinear liquid sloshing in non-rectangular containers. Additionally, installing obstacles inside the container can significantly alter the liquid sloshing response, with vertical dimension changes exerting a greater influence than horizontal ones.