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Shih, Chao-Feng (National Taiwan Ocean University / Taiwan International Ports Corporation, Ltd.) | Chen, Yung-Wei (National Taiwan Ocean University) | Soon, Shih-Ping (National Taiwan Ocean University) | Ho, Sheng-Yu (National Taiwan Ocean University)
In this paper, we use the multiple scale Trefftz method (MSTM) combined with the Lie group scheme to simulate the two-dimensional nonlinear sloshing problem. When considering the effects of baffles for the nonlinear sloshing phenomena, the conventional Trefftz method (CTM) may encounter numerical instabilities, degenerate scale, and numerical dissipation problem. In order to eliminate the high-order numerical oscillations and noise disturbances on the boundary, the vector regularization method (VRM) and the multiple scale characteristic lengths (CLs) are applied. At the same time, they can also overcome the degenerate scale problem. Additionally, in order to increase the numerical accuracy at the initial-boundary value problem, we introduce the weighting factors based on the Lie group scheme into the linear system to avoid high numerical dissipation and reduce the numbers of iterations. Comparing with the solutions in the previous literature, the present scheme is efficient and accurate in simulating nonlinear sloshing problem of the two-dimensional tank with baffles. Hence, the method developed here is a simple and stable way to cope with the nonlinear sloshing problem with baffles.
Liquid sloshing in a moving container has been studied in various engineering problems, such as liquid propellant launch vehicles (Mohammad et al., 2011), oil surges in large tanks due to long-term intense ground motion (Tetsuya and Takashi, 2013), and shock surges in nuclear fuel pools (Eswaran and Reddy, 2016). Besides, the sloshing effect in the ship’s ballast tanks may cause it to experience large rolling moments, even losing dynamic stability and overturning (Przemysáaw et al., 2012). Also, if the forcing frequency coincides with the natural sloshing frequency, the high dynamic pressures, by reason of resonance, may damage the tank walls and may even create moments that affect the stability of the vehicle. Therefore, the hydrodynamics of sloshing for the vehicle is important and requires understanding sloshing dynamics phenomena.
Chen, Yung-Wei (National Taiwan Ocean University) | Shih, Chao-Feng (National Taiwan Ocean University) | Liu, Yu-Chen (National Taiwan Ocean University) | Soon, Shih-Ping (National Taiwan Ocean University)
This paper presents an equal-norm multiple-scale Trefftz method (MSTM) associated with the group-preserving schemes (GPS) to tackle some difficulties in nonlinear sloshing behaviors. The MSTM combined with the vector regularization method is first adopted to eliminate the higher-order numerical oscillation phenomena and noisy dissipation in the boundary value problem. Then, the weighting factors of initial and boundary value problems are introduced into the linear system to prevent the elevation from vanishing without iterative computational controlled volume. More important, the explicit scheme, based on the GL (n, R), and the implicit scheme can be combined to reduce iteration number and increase computational efficiency. A comparison of the results shows that the proposed approach is better than previously reported methods.
Sloshing of liquid in tanks has received considerable attention from many researchers in related engineering fields. The problem arises because excessive sloshing of the confined liquid can strongly damage the structure or the loads induced by sloshing, which may seriously modify the dynamics of the vehicle that supports the tanks—for example, fuel sloshing in liquid propellant launch vehicles (Lu et al., 2015), oil oscillations in large storage tanks as a result of long-period strong ground motions (Hashimoto et al., 2017), and sloshing in nuclear fuel pools owing to earthquakes (Eswaran and Reddy, 2016). Besides, sloshing effects in the ballast tanks of a ship may cause it to experience large rolling moments and eventually capsize because of loss of dynamic stability (Krata, 2013; Sanapala et al., 2018). Also, if the forcing frequency coincides with the natural sloshing frequency, the high dynamic pressures, by reason of resonance, may damage the tank walls. Thus, accurate prediction of sloshing behaviors in tanks driven by external forces is very critical for successful structural design and reducing impacts on vehicle maneuvering.
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)
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.
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.