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Continuous Finite Element Implementation

Continuous finite element implementation is achieved by working directly from the variational indicator. In order to derive the virtual work statements to be discretized, the variations of Equation (4.10) with respect to $\phi$ and un are taken, thus

  
$\displaystyle \delta \phi:$ $\displaystyle \;\;\;\; \rho_f \int_{\Omega_f} \mbox{{\boldmath$\bigtriangledown... ... d \Omega_f - \rho_f \int_{\Gamma_f} \delta \phi \; \dot{u}_n \; d \Gamma_f = 0$(167)
$\displaystyle \delta u_n:$ $\displaystyle \;\;\;\; \int_{\Gamma_s} P \delta u_n \; d \Gamma_s = 0$(168)

Equation (4.42) may be rewritten as

\begin{displaymath}\delta \phi:\;\;\;\; \rho_f \int_{\Omega_f} \mbox{{\boldmath$... ...ac{\partial \phi}{\partial {\bf n}} \right) \; d \Gamma_f = 0 \end{displaymath}(169)

Since the virtual variables are arbitrary, the Euler equations that govern the liquid behavior are
   
$\displaystyle \mbox{Continuity Condition:}$ $\displaystyle \;\;\; \bigtriangledown^2 \phi=0 \;\;\; \mbox{on $\Omega_f$ }$(170)
$\displaystyle \mbox{Kinematic Condition:}$ $\displaystyle \;\;\; \dot{u}_n =\frac{\partial \phi}{\partial {\bf n}} \;\;\; \mbox{on $\Gamma_f$ }$(171)
$\displaystyle \mbox{Dynamic Condition:}$ $\displaystyle \;\;\; P=0 \;\;\; \mbox{on $\Gamma_s$ }$(172)

Equation (4.46) represents the kinematic condition at the liquid-structure interface $\Gamma_w$ and at the free surface $\Gamma_s$. On the other hand, Equation (4.47) represents the dynamic condition only at the free surface $\Gamma_s$ because un is considered as unknown only at the free surface.

Linearization of Equations (4.42) and (4.43) is achieved by assuming that the free surface elevation h is equal to the normal displacement of the free surface un and ignoring the term $\bigtriangledown$$^2 \phi$ in Equation (4.12), this yields to

 
$\displaystyle \left[ \begin{array}{ccc} 0 & {\bf M}_f^j \\ {\bf M}_f^j & 0 \en... ...\begin{array}{c} \delta \dot{\Phi} \\ \delta {\bf\dot{h}} \end{array} \right\}$+$\displaystyle \left[ \begin{array}{ccc} 0& 0 \\ 0& g {\bf M}_f^j \end{array} \... ...right\} = \left\{ \begin{array}{c} {\bf R}_k \\ {\bf R}_d \end{array} \right\}$(173)
$\displaystyle {\bf R}_d$=$\displaystyle {\bf M}_f^j \left\{ \frac{\partial \phi}{\partial t} +\frac{\bigtriangledown^2 \phi}{2} + gh\right\}$(174)
$\displaystyle {\bf R}_k$=$\displaystyle {\bf M}_f^j {\bf v}_n - \rho_f \int_{\Omega} {\bf B}_f^T {\bf V}_f \; d\Omega$(175)

where j is the free surface edge number, ${\bf V}_f$ is the liquid velocity vector and ${\bf v}_n$is the normal velocity vector at free surface nodes. The normal velocity at a free surface node i is computed as

\begin{displaymath}v_{ni} = {\bf n}_i^T \left\{ \begin{array}{c} V_x \\ \dot{h }\\ V_z \end{array} \right\}_i \end{displaymath}(176)

where ${\bf n}_i$ is the normal vector to the free surface at the node, and Vx and Vz are the velocity of free surface node i in the x and z directions due to the mesh speed, respectively.

next up previous contents
Next: Nonlinear Liquid Sloshing in Up: Nonlinear Liquid Sloshing in Previous: Discrete Finite Element Implementation
A. Zeiny
2000-09-06