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Table of Symbols

AiTributary area of a liquid free surface node projected in the horizontal plane.
${\bf B}$ or ${\bf B}_o$Linear strain-displacement matrix.
${\bf B}_l$Nonlinear strain-displacement matrix.
${\bf C}_s$Solid domain damping matrix.
${\bf C}_{fs}$Liquid-solid interaction matrix.
${\bf C}_{sf}$Solid-liquid interaction matrix.
${\bf C}_{\phi h}$Liquid-free surface interaction matrix.
${\bf E}$Constitutive matrix.
${\bf F}_{ext}$External force/discharge vector.
${\bf F}_{int}$Internal force/discharge vector.
${\bf F}_{I}$ and ${\bf F}_{E}$Interface and external body forces, respectively.
Fn and FtNormal and tangential contact forces, respectively.
gAcceleration of gravity.
${\bf G}$Ground motion vector.
${\bf h}$Vector of liquid free surface heights.
HTank height.
${\bf J}$Element Jacobian matrix.
${\bf K}_{s}$, ${\bf K}_{f}$ and ${\bf K}_{h}$Solid, liquid and free surface element stiffness matrices, respectively.
${\bf K}_{\lambda}$Contact/constraint matrix.
lxProjected width of liquid element.
L1, L2 and L3Nondimensional area coordinates.
${\bf n}$Unit vector normal to the element face.
NiShape function at node i.
${\bf M}^j_f$Liquid element face mass matrix.
${\bf M}_s$Structural mass matrix.
PLiquid total pressure.
${\bf P}$Contact penetration vector.
${\bf q}$Lamina transformation matrix.
${\bf r}$Nodal position vector.
RRadius of a cylindrical tank or base width of a rectangular tank.
${\bf R}_d$ and ${\bf R}_k$Residual vectors of the dynamic and kinematic conditions, respectively.
${\bf R}_f$Residual vector of liquid domain nodes.
TaTributary surface area of a solid domain node.
${\bf u}$, ${\bf\dot{u}}$ and ${\bf\ddot{u}}$Nodal displacement, velocity and acceleration vectors, respectively.
${\bf u'}$Transformed displacement vector.
${\bf v}_3$Fiber direction.
vn and $\dot{u}_n$Liquid and solid normal velocity at a solid-liquid interface, respectively.
${\bf V}$Velocity of a fluid particle.
${\bf V}_{r}$Velocity of a fluid particle relative the fluid mesh.
${\bf x}_i^t$ and ${\bf x}_i^b$Top and bottom nodal coordinate vectors, respectively.
x', y' and z'Transformed lamina coordinate system.
$\Gamma_f$Liquid domain boundary.
$\Gamma_s$Free surface boundary.
$\Gamma_w$Liquid flexible boundary.
$\theta$Nodal rotational vector.
$\lambda$Lagrange multipliers vector.
$\nu_x$ and $\mu$Free surface damping parameters.
$\Pi$Energy functional.
$\Omega_s$ and $\Omega_f$Solid and liquid domains, respectively.
$\phi$ and $\Phi$Velocity potential function and vector, respectively.
$\bigtriangledown^2$Laplacian differential operator.
$\bigtriangledown$Gradient differential operator.
$\xi$, $\eta$ and $\zeta$Element local non-dimensional coordinates.
$\rho_s$ and $\rho_f$Solid and liquid mass densities, respectively.
$\sigma$Stress and strain vectors, respectively.

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A. Zeiny