3.2.4.15. Stress Tensor Projection Equation

A projection equation is defined as an equation where a derived quantity at the interior Gauss points is evaluated and projected to be a solution unknown at the nodal points. The stress tensor projection equation projects the momentum stresses, tau, without the pressure term, to the nodal points. Projecting the momentum stress smooths out the momentum stress tensor and allows for a dot product to be carried out on the projected field, which is needed for least squares stabilization schemes. The solution variable, ${\tau}_{ab}$ , is calculated from (3.73).

(3.73) $\symRes_V^i = - \underbrace{\int\limits_\Vol ({\tau}_{ab} - {src\_\tau}_{ab}) \phi^i\dV}_\mathrm{DEF} = 0$

${\tau}_{ab}$ is a tensor variable. For 2D problems, $ab$ stands for XX, XY, YX, and YY. For 3D problems $ab$ stands for XX, XY, XZ, YX, YY, YZ, ZX, ZY, and ZZ. The source term in the equation ${src_\tau}_{ab}$ refers to the momentum stress without the pressure diagonal term.

${src\_\tau}_{xx} = 2 \mu \frac{{d}{u}}{dx} - \frac{2}{3} (\mu - \lambda) (\div\vector{v}) \\ {src\_\tau}_{yy} = 2 \mu \frac{{d}{v}}{dy} - \frac{2}{3} (\mu - \lambda) (\div\vector{v}) \\ {src\_\tau}_{zz} = 2 \mu \frac{{d}{w}}{dz} - \frac{2}{3} (\mu - \lambda) (\div\vector{v}) \\ {src\_\tau}_{xy} = {src\_\tau}_{yx} = \mu (\frac{{d}{u}}{dy} + \frac{{d}{v}}{dx} ) \\ {src\_\tau}_{xz} = {src\_\tau}_{zx} = \mu (\frac{{d}{u}}{dz} + \frac{{d}{w}}{dx} ) \\ {src\_\tau}_{yz} = {src\_\tau}_{zy} = \mu (\frac{{d}{v}}{dz} + \frac{{d}{w}}{dy} )$