Three‐dimensional sloshing: A consistent finite element approach

Abstract

This paper presents a new computational methodology based on Legendre's polynomials to predict the slosh and acoustic motion in nearly incompressible fluids in both rigid and flexible structures with free surface. Here, we have used a finite element formulation based on Lagrangian frame of reference to model the fluid motion derived using Hamiltonian equation of the fluid system. We formulated three hexahedral finite elements based on strain fields expressed in terms of extended Legendre's polynomials. Sloshing and acoustic motion of liquid is investigated using these newly formulated elements and inf-sup test is performed on these new elements to check the performance of these elements in modeling sloshing under two severe constraints, namely incompressibility and irrotationality. Comparisons of slosh and acoustic frequencies, and mode shapes with exact solutions are given. Dynamic analysis with earthquake and harmonic kind of forcing function is carried out to validate the formulated hexahedral elements to analyze the sloshing response. Numerical results obtained with these new finite elements, and with the present finite element formulation of the mathematical model agree well with the exact solution and as well as with published experimental literature.In this paper, we study the behaviour of the family of 3-D Lagrangian fluid elements for modeling slosh and acoustic motion, and suggest methods to overcome these two severe constraints.The most common methods of solving fluid-structure interaction problems are the method based on mixed formulation and the method based on arbitrary Lagrangian-Eulerian formulation [4]. These procedures result in unsymmetric matrices which require special solvers. On the other hand, solving fluid-structure interaction problems using fluid elements formulated using a Lagrangian frame of reference has the special advantage of no explicit coupling. Enforcement of equilibrium at the fluid-solid interface ensures proper coupling. This can be accomplished by simple assembly of stiffness matrices of fluid and solid domains. The final matrices will be symmetric and banded and conventional banded or skyline solvers are sufficient for efficient and economic solution. Therefore, the conventional finite element solvers used for structural analysis can be easily modified to perform fluid-structure interaction analysis.Several researchers have employed displacement-based Lagrangian fluid elements. Belytschko have used this approach extensively in linear and nonlinear transient fluid-structure interaction problems [7-10]. Gladwell and Zimmerman [11] and Gladwell [12] present a variational formulation of acousto-structural problems using displacements or pressures as variables. However, although various applications have been reported using displacement-based fluid elements, there are still some difficult questions that need to be addressed regarding the applicability of the formulation to various analysis. As noted earlier by Kiefling and Feng [13], and Hamdi and Ousset [14], the displacement-b...