**Next:** Acknowledgment **Up:** Title Page **Previous:** List of Tables

**Table of Symbols**

A_{i} | Tributary area of a liquid free surface node projected in the horizontal plane. |

or | Linear strain-displacement matrix. |

Nonlinear strain-displacement matrix. | |

Solid domain damping matrix. | |

Liquid-solid interaction matrix. | |

Solid-liquid interaction matrix. | |

Liquid-free surface interaction matrix. | |

Constitutive matrix. | |

External force/discharge vector. | |

Internal force/discharge vector. | |

and | Interface and external body forces, respectively. |

F_{n} and F_{t} | Normal and tangential contact forces, respectively. |

g | Acceleration of gravity. |

Ground motion vector. | |

Vector of liquid free surface heights. | |

H | Tank height. |

Element Jacobian matrix. | |

, and | Solid, liquid and free surface element stiffness matrices, respectively. |

Contact/constraint matrix. | |

l_{x} | Projected width of liquid element. |

L_{1}, L_{2} and L_{3} | Nondimensional area coordinates. |

Unit vector normal to the element face. | |

N_{i} | Shape function at node i. |

Liquid element face mass matrix. | |

Structural mass matrix. | |

P | Liquid total pressure. |

Contact penetration vector. | |

Lamina transformation matrix. | |

Nodal position vector. |

R | Radius of a cylindrical tank or base width of a rectangular tank. |

and | Residual vectors of the dynamic and kinematic conditions, respectively. |

Residual vector of liquid domain nodes. | |

T_{a} | Tributary surface area of a solid domain node. |

, and | Nodal displacement, velocity and acceleration vectors, respectively. |

Transformed displacement vector. | |

Fiber direction. | |

v_{n} and | Liquid and solid normal velocity at a solid-liquid interface, respectively. |

Velocity of a fluid particle. | |

Velocity of a fluid particle relative the fluid mesh. | |

and | Top and bottom nodal coordinate vectors, respectively. |

x', y' and z' | Transformed lamina coordinate system. |

Liquid domain boundary. | |

Free surface boundary. | |

Liquid flexible boundary. | |

Nodal rotational vector. | |

Lagrange multipliers vector. | |

and | Free surface damping parameters. |

Energy functional. | |

and | Solid and liquid domains, respectively. |

and | Velocity potential function and vector, respectively. |

Laplacian differential operator. | |

Gradient differential operator. | |

, and | Element local non-dimensional coordinates. |

and | Solid and liquid mass densities, respectively. |

Stress and strain vectors, respectively. |