Application of the Lie-Group Scheme in Euclidean Space Solving Nonlinear Sloshing Behaviors

Soon, Shin-Ping (National Taiwan Ocean University) | Shih, Chao-Feng (National Taiwan Ocean University, Taiwan International Ports Corporation, Ltd.) | Chen, Yung-Wei (National Taiwan Ocean University) | Liu, Yu-Chen (National Taiwan Ocean University)

OnePetro 

ABSTRACT

An equal norm multiple scale Trefftz method (MSTM) associated with the Lie-group scheme GL(n, R) in Euclidean space is developed to describe two dimensional nonlinear sloshing behaviors. In this paper, the explicit and implicit GL(n, R) in the Euclidean space are used for time integration and the results in terms of computational efficiency and accuracy are very good. The MSTM combined with the vector regularization method (VRM) is adopted the first time to eliminate the phenomena of higher-order numerical oscillation and noisy dissipation. The proposed method in this paper can overcome the boundary noisy disturbance and improve the stability and accuracy of the sloshing problems. Numerical scheme is developed and verified by benchmark tests. Different shapes of the fluid tank are simulated with various excitation frequencies. The occurring waves are successfully modeled and the results will be discussed later in detail. Comparisons of the results with other methods shows that the proposed method in this paper indeed does a better job on both accuracy and running time.

INTRODUCTION

In recent years, the application of SPH method by Monaghan [Kim Y, 2001], Ma[Zhang T, etc. 2016], Jan [Monaghan JJ, 1994], respectively, the application of SPH method, MLPG[Zhang T, etc. 2016], respectively, the simulations of sloshing behaviors exist singular problems of integrals and slow in convergence due to their finite element and boundary element methods. Local Radial Basis Function Collocation Method (LRBFCM) has also been used effectively to simulate the sloshing phenomenon, such as Fan and other scholars [Vaughan GL, etc. 2008]. In this paper, the Trefftz method is used to solve the sloshing problem. Trefftz method [Ma QW, 2005] was first proposed in 1926. Trefftz method using T-complete function as a base function to meet the problem of governing equations. In 2004, Kita et al. [Ali A, etc. 2005] first applied the Trefftz method to the simulation of the sloshing problem. When applying the Trefftz method with no singular sources, sufficient constraint equations should be established to increase the boundary discrete points in order to improve the simulation accuracy. But the high order base functions will cause numerical instability. Liu [Liu CS, etc. 2009] proposed modified Trefftz method (MTM) to introduce the feature length in the T-complete base function to improve the phenomenon of numerical instability, Chen [Chen YW, 2009, 2010, 2012] and other scholars using the modified Trefftz method And Geometric Multiple Scale Trefftz Method (MSTM) to solve the sloshing problem. The concept of dissipation factor and control volume is revised to improve the accuracy of the solution. In 2016, MTM and VRM were proposed to overcome Border interference. In the part of solving the initial value problem of sloshing problem, this paper will use the preserving group algorithm to carry on the operation, and the preserving group algorithm is generalized through the group concept. Some scholars of modern mathematicians Hall [Li ZC, etc 2013], and American scholar Lang [Ramachandran PA 2002] also discussed Lie-group and Lie algebra. As the development of group theory is maturing, matures, many physical problems are solved, such as coupling problem, boundary value problem and ordinary differential equation problem. Liu [Chen CS, etc 2006] introduced nonlinear dynamic system into augmented dynamic system in 2001 (Minkowski space) under the deduction of Lorentz group. And in 2013 [Jin BA, 2004] derived Lie-group differential algebraic equation method for solving the above problems. Lie-group differential algebraic equation method is a nonlinear differential algebraic equation, adding generalized linear group structure, so that it is converted to initial value problem solving. Through the developed method, simple formulae are derived to deal with complex problems. In this paper, we will use the explicit and implicit Lie-groups method in the Euclidean space for time integration. In this paper, the computational efficiency and accuracy are very good when the Lie method is used in the Euclidean space. The weighting factor calculated by the group preserving algorithm is introduced into the boundary value problem to correct the numerical error of the boundary value problem.