| Matrix or Vector | Shape Functions | Integration Points |
|---|---|---|
| Stiffness and Damping Matrices; and Thermal Load Vector | Equation 11–217, Equation 11–218, and Equation 11–219 |
1 x 1 x 1 for bulk strain effects |
| Mass Matrix | Same as stiffness matrix. Matrix is diagonalized as described in Lumped Matrices | 2 x 2 x 2 |
| Pressure Load Vector | Same as stiffness matrix, specialized to the face | 2 x 2 |
| Load Type | Distribution |
|---|---|
| Element Temperature | Average of the 8 nodal temperatures is used throughout element |
| Nodal Temperature | Average of the 8 nodal temperatures is used throughout element |
| Pressure | Bilinear across each face |
Structures describes the derivation of element matrices and load vectors.
This element does not generate a consistent mass matrix; only the lumped mass matrix is available.
Rather than Equation 2–3, the stress-strain relationships used to develop the stiffness matrix and thermal load vector are:
 | (1–116) |
where:
 |
| α = thermal coefficient of expansion (input as ALPX on MP command) |
| ΔT = change of temperature from reference temperature |
| K = fluid elastic (bulk) modulus (input as EX on MP command) |
| P = pressure |
| γ = shear strain |
| S = K x 10-9 (arbitrarily small number to give element some shear stability) |
| τ = shear stress |
| Ri = rotation about axis i |
| B = K x 10-9 (arbitrarily small number to give element some rotational stability) |
| Mi = twisting force about axis i |
A damping matrix is also developed based on:
 | (1–117) |
where:
| η = viscosity (input as VISC on MP command) |
| c = .00001*η |
and the (
) represents differentiation
with respect to time.
A lumped mass matrix is developed, based on the density (input as DENS on MP command).
The free surface is handled with an additional special spring effect. The necessity of these springs can be seen by studying a U-Tube, as shown in Figure 1.12: U-Tube with Fluid.
Note that if the left side is pushed down a distance of Δh, the displaced fluid mass is:
 | (1–118) |
where:
| MD = mass of displaced fluid |
| Δh = distance fluid surface has moved |
| A = cross-sectional area of U-Tube |
| ρ = fluid density |
Then, the force required to hold the fluid in place is
 | (1–119) |
where:
| FD = force required to hold the fluid in place |
| g = acceleration due to gravity (input on ACEL command) |
Finally, the stiffness at the surface is the force divided by the distance, or
 | (1–120) |
This expression is generalized to be:
 | (1–121) |
where:
| AF = area of the face of the element |
| gi = acceleration in the i direction |
| Ci = ith component of the normal to the face of the element |
This results in adding springs from each node to ground, with the spring constants being positive on the top of the element, and negative on the bottom. For an interior node, positive and negative effects cancel out and, at the bottom where the boundary must be fixed to keep the fluid from leaking out, the negative spring has no effect. If KEYOPT(2) = 1, positive springs are added only to faces located at z = 0.0.
The surface springs tend to retard the hydrostatic motions of the element from their correct values. The hydrodynamic motions are not changed. From the definition of bulk modulus,
 | (1–122) |
where:
| us = vertical motion of a static column of fluid (unit cross-sectional area) |
| H = height of fluid column |
| P = pressure at any point |
| z = distance from free surface |
The pressure is normally defined as:
 | (1–123) |
But this pressure effect is reduced by the presence of the surface springs, so that
 | (1–124) |
Combining Equation 1–122 and Equation 1–124 and integrating,
 | (1–125) |
or
 | (1–126) |
If there were no surface springs,
 | (1–127) |
Thus the error for hydrostatic effects is the departure from 1.0 of the factor (1 / (1+Hρg/K)), which is normally quite small.
The 1 x 1 x 1 integration rule is used to permit the element to "bend" without the bulk modulus resistance being mobilized, i.e.
While this motion is permitted, other motions in a static problem often result, which can be thought of as energy-free eddy currents. For this reason, small shear and rotational resistances are built in, as indicated in Equation 1–116.