Matrix or Vector | Shape Functions | Integration Points |
---|---|---|
Conductivity Matrix and Heat Generation Load Vector | Equation 11–224 | 2 x 2 x 2 |
Specific Heat Matrix | Equation 11–224. Matrix is diagonalized as described in Lumped Matrices | Same as conductivity matrix |
Convection Surface Matrix and Load Vector | Equation 11–224 specialized to the face | 2 x 2 |
Derivation of Heat Flow Matrices has a complete derivation of the matrices and load vectors of a general thermal analysis element. Mass transport is discussed in PLANE55 - 2-D Thermal Solid.
An option (KEYOPT(7) = 1) is available to convert SOLID70 to a nonlinear steady-state fluid flow element. Pressure is the variable rather than temperature. From Equation 6–22, the element conductivity matrix is:
(13–134) |
[B] is defined by Equation 6–22 and for this option, [D] is defined as:
(13–135) |
where:
= absolute permeability of the porous medium in the x direction (input as KXX on MP command) |
ρ = mass density of the fluid (input as DENS on MP command) |
μ = viscosity of the fluid (input as VISC on MP command) |
β = visco-inertial parameter of the fluid (input as C on MP command) |
S = seepage velocity (at centroid from previous iteration, defined below) |
α = empirical exponent on S (input as MU on MP command) |
For this option, no “specific heat” matrix or “heat generation” load vector is computed.
The pressure gradient components are computed by:
(13–136) |
where:
= pressure gradient in the x-direction (output as PRESSURE GRADIENT (X)) |
{T_{e}} = vector of element temperatures (pressures) |
The pressure gradient is computed from:
(13–137) |
where:
g^{p} = total pressure gradient (output as PRESSURE GRADIENT (TOTAL)) |
The mass flux components are:
(13–138) |
The vector sum of the mass flux components is:
(13–139) |
where:
f = mass flux (output as MASS FLUX) |
The fluid velocity components are:
(13–140) |
where:
S_{x} = fluid velocity in the x-direction (output as FLUID VELOCITY (X)) |
and the maximum fluid velocity is:
(13–141) |
where:
S = total fluid velocity (output as FLUID VELOCITY (TOTAL)) |