liquid sloshing is defined to be the free surface gravity waves caused by an excitation to a liquid domain. Figure (4.3) shows the geometry of the problem of liquid sloshing in a rigid tank. In the special case of rigid tank, the translational motion of the liquid boundaries are directly proportional to the ground excitation.

The normal velocity of the boundaries is then given by

where is the normal to the liquid boundary,

(154) |

The strong form of the boundary value problem that represent the nonlinear liquid sloshing in a rigid tank is stated as follows:

Given , the initial free surface , and the initial conditions of the potential function , find and such that

where is the free surface boundary, is liquid boundary in contact with the tank wall or base plate and

As a result of the simplified assumptions, lack of viscosity sometimes causes undesirable contribution from the high frequency components in the numerical solution of the problem. This contribution is undesirable because of the high frequency modes in the solution that are poorly represented in the discritized system. Any wave that has a length shorter than the element width is not represented adequately due to the limited order of the shape functions. As a result, dispersion error may develop in the solution of some problems. This occurs, sometimes, when the liquid is in the resonance zone or when the excitation level is relatively high. Numerical dissipation may be used then to damp out the high frequency wave components propagating near the free surface. A numerical dissipation term was suggested by Chen [32] for the two-dimensional solution of the problem. He reported that this term has a little effect on the low frequency modes that govern the solution. After incorporating the numerical dissipation, Equation (4.35) may be rewritten in the two-dimensional case as

where is the damping parameters given as function of the mesh density,

(162) |

where

In case of large amplitude surface waves, it is desirable to update the mesh to follow the liquid boundaries and to avoid distorted elements. In such a case, the time derivatives of any spatial function should be modified to account for mesh speed according to the following relation

(163) |

where is the mesh speed and